INVESTIGATION OF THE STABILITY OF EQUILIBRIUM POINTS IN THE
RELATIVISTIC RESTRICTED THREE-BODY PROBLEM WITH
PERTURBATIONS
BY
BELLO NAKONE
DEPARTMENT OF MATHEMATICS
FACULTY OF SCIENCE
AHMADU BELLO UNIVERSITY
ZARIA, NIGERIA
SEPTEMBER, 2016
i
INVESTIGATION OF THE STABILITY OF EQUILIBRIUM POINTS IN THE
RELATIVISTIC RESTRICTED THREE-BODY PROBLEM WITH
PERTURBATIONS
BY
Bello NAKONE
B.Sc. (U.D.U.S.), M.Sc. (B.U.K.), M. TECH (F.U.T. MINNA)
Ph.D/SCIE/2600/2010-2011
A THESIS SUBMITTED TO THE
SCHOOL OF POSTGRADUATE STUDIES,
AHMADU BELLO UNIVERSITY
ZARIA, NIGERIA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD
OF DOCTOR OF PHILOSOPHY IN MATHEMATICS
SEPTEMBER, 2016
ii
DECLARATION
I declare that the work in this thesis titled INVESTIGATION OF THE STABILITY
OF EQUILIBRIUM POINTS IN THE RELATIVISTIC RESTRICTED THREEBODY PROBLEM WITH PERTURBATIONS has been performed by me under the
supervision of Prof. J. Singh, Prof. B. K. Jha, and Dr. A. O. Ajibade in the Department
of Mathematics, Ahmadu Bello University, Zaria. The information derived from the
literature has been duly acknowledged in the text and a list of references provided. No
part of this thesis was previously presented for another degree or diploma at any other
institution.
Bello Nakone
Name of student
_________________
Signature
iii
__________________
Date
CERTIFICATION
This Thesis titled INVESTIGATION OF THE STABILITY OF EQUILIBRIUM POINTS IN
THE RELATIVISTIC RESTRICTED THREE-BODY PROBLEM WITH PERTURBATIONS
by Bello NAKONE meets the regulation governing the award of the degree of Doctor
of Philosophy in Mathematics of Ahmadu Bello University, and is approved for its‟
contribution to knowledge and literary presentation.
__________________
_______________
Prof. J. Singh
Date
Chairman, Supervisory Committee
_______________
_______________
Prof. B. K. Jha
Date
Member, Supervisory Committee
__________________
_______________
Dr. A. O. Ajibade
Date
Member, Supervisory Committee
__________________
_______________
Prof. E. J. D. Garba
Date
External Examiner
_________________
_______________
Prof. A. A. Tijjani
Date
Head of Department
__________________
_______________
Prof. Kabiru Bala
Date
Dean, School of Postgraduate Studies
iv
ACKNOWLEDGMENT
All praises and gratitude are for Almighty Allah, who has the monopoly of all
knowledge. May the peace and blessings of Allah be on the Noble Prophet Mohammed
(S.A.W.).
I heartily acknowledge the patience and prayers of my mother and my family.
I would like to thank my supervisor Prof. J. Singh for his dedication and tireless efforts.
His patience, guidance, support and advice. I am also grateful for the privilege of
continuing to learn from him.
Next, I would like to thank Prof. B. K. Jha, Dr. A. O Ajibade as members of my
supervisory committee. My profound gratitude goes to Prof. A. K. Adamu (Dean of
faculty of science) and Prof. A. A. Tijjani (Head of Department of Mathematics) and
Prof. B. Sani (immediate past Head of Department of Mathematics).
Additionally, I would like to express my appreciation to Post-Graduate coordinator, Dr.
A. Yahaya, Seminar coordinator, Dr. A. M. Ibrahim and assistant seminar coordinator,
Mal. A. T. Imam for their patience in coordinating Post-Graduate activities in the
Department. In addition, I would like also to thank all staff members of the Department
such as Prof. D. Singh, Prof. S. Junaidu, Dr. H. M. Jibril, Dr. H. G. Dikko, Dr. A. A.
Obiniyi, Dr. I. A. Fulatan, Dr. Y. M. Baraya, Dr. A. J. Alkali, Dr. A. Mohammed, Dr.
A. D. F. Kana, Dr. I. A. Baroon, to mention but a few.
I must give special thanks to Dr. (Mrs) Aishetu Umar for her patience with us and her
help in the understanding of many concepts in writing this thesis. I am very thankful to
Mal. Aminu Abubakar Hussain, Mal. Amuda Tajudeen and Mr. Aina Babatunde for
their contribution in the preparation of this thesis.
v
I am also grateful to Dr. A. AbdulRazaq and my colleagues, such as Dr. L. S. Oni, Dr. J.
J. Taura, Mr. A. E. Vincent and Mrs B. J. Mrumum.
Additionally, I would like to express my appreciation to each and every present and past
member of my research group.
It is pleasure to express my sincere gratitude to Prof. S. U. Gulumbe, Prof. A. U. Ali,
Prof. M. O. Ibrahim and all the academic and non academic staff in the Department of
Mathematics of Usmanu Danfodiyo University, Sokoto.
My special thanks go to Mal. N. S. Dauran, Dr. A. Mustapha, Mal. A. B. Zoramawa,
Dr. I. A. Garba, Mal. A. Roko, Mal. Alh. B. Buhari, Mal. F. R. Tambuwal, Engr. Salou
Moussa, Alh. Mukthar Garba, Dr. B. M. Yerima and my neighbor Mal. Sulaiman
Taofiq for the services they have rendered to me.
Some thanks are also given to Prof. V. S. Kalantonis, Prof. E. A. Perdios, Prof. M. F.
Roy, Prof. D. Boucher, Prof. D. B. Meade, Prof. P. Woolf , Prof. S. M. Kopeikin, Prof.
G. I. Balogun, Dr. E. I. Abouelmagd and Dr. Bala Yabo for their assistance.
My gratitude also goes to the authorities of IMSP (Port-Novo) and CIPMA (Cotonou)
both in Benin Republic for inviting me to many of the conferences that they have
organized and which they were fruitful to me.
My sincere gratitude goes also to the authority of Usmanu Danfodiyo University,
Sokoto for providing me the opportunity, as well as making it possible for me to
undertake this research.
Finally, I thank all those who have supported me in the
preparation of this thesis who are too numerous to be mentioned.
vi
DEDICATION
To my late father Nakone Mayaki and my late grand mother Tacorgo Koussougou.
vii
ABSTRACT
In the work by Bhatnagar and Hallan (1998), linear stability of the relativistic triangular
L4 and L5 points was studied and it was shown that these points were unstable for the
whole range 0   
1
despite the well-known fact that the non-relativistic L 4 and L5
2
are stable for   0 , where 0 is the Routh critical mass ratio. The same problem was
later investigated by Douskos and Perdios (2002) and Ahmed et al. (2006) and they
obtained two different ranges of mass ratios in which the relativistic triangular points
are linearly stable in contradiction with the result of Bhatnagar and Hallan (1998). In
this thesis we reconsider and generalize the problem investigated by these authors in
that perturbations in the Coriolis and centrifugal forces, radiation pressure, oblateness
and triaxiality factors of the primaries have been considered in our investigation. The
locations of equilibrium points are obtained and their stability are analyzed by using
variational method and Lyapunov‟s criteria. The triangular points of the relativistic
three-body problem (R3BP) are studied from various aspects of perturbations such as
oblateness, triaxiality and radiation pressure of the primaries as well as the small
perturbations in the centrifugal and Coriolis forces. It is found that the locations of the
triangular points are affected by the asphericity of the primaries, the relativistic terms
and a small change in the centrifugal force. It is also found that the triangular points are
stable for 0     c and unstable for  c   
1
, where  c is the critical mass
2
parameter depending on the perturbation parameters and relativistic terms. It is further
found that the Coriolis force has stabilizing tendency, while the centrifugal force,
radiation pressure forces, oblateness, triaxiality of the primaries and relativistic terms
have destabilizing effects. The motion of an infinitesimal mass near the collinear
equilibrium points when the smaller primary is a triaxial body is also studied. It is
observed that the positions of the collinear points are affected by the relativistic and
triaxiality factors. The collinear points are found to remain unstable. Numerical studies
in this connection with the Sun-Earth, Sun-Pluto and Earth-Moon systems have been
carried out to show the relativistic and triaxiality effects. The motion of an infinitesimal
mass near the collinear equilibrium points when the smaller primary is oblate is also
investigated. The collinear points are found to be unstable. A numerical exploration in
this connection, with some members of our solar system reveals that the locations of the
viii
collinear points L1 , L2
are affected prominently by the relativistic factor in the absence
of oblateness and they are also affected significantly by the oblateness factor in the
absence of relativistic terms. It is also found that in most of the cases, the position
of
L3 is negligibly affected by the relativistic and oblateness factors. More specifically, all
parameters involved have no effect on the position of L3 of the Sun-Mars system. The
results show that the oblateness and relativistic factors have the same but separate effect
on the position of L1 of the Sun-Uranus system and have also the same effect on the
position of L1 of the Sun-Neptune system. It is also found that in the presence of
relativistic terms, the effect of oblateness on the Sun-Planet pairs does not show
physically. Also, the frequencies of the long and short orbit of the periodic motion,
eccentricities, axes and the orientation of the orbits around the stable triangular points
when the bigger primary is triaxial are determined and found to be affected by the
triaxiality and relativistic effects. The results of this study generalise the classical
relativistic restricted three-body problem (R3BP) and the results of Douskos and
Perdios (2002) can be deduced from this study while the present results differ with the
results of Bhatnagar and Hallan (1998) and differ also with the results of Ahmed et al.
(2006).
ix
TABLE OF CONTENTS
Cover Page ...................................................................................... Error! Bookmark not defined.
Fly Leaf ........................................................................................... Error! Bookmark not defined.
Title Page ........................................................................................ Error! Bookmark not defined.
Declaration ....................................................................................................................................iii
Certification .................................................................................................................................. iv
Acknowledgment ...........................................................................................................................v
Dedication .................................................................................................................................... vii
Abstract ....................................................................................................................................... viii
Table of Contents ...........................................................................................................................x
List of Figures ............................................................................................................................. xiii
List of Tables .............................................................................................................................. xiv
List of Symbols ............................................................................................................................ xv
CHAPTER ONE ......................................................................................................................... 1
INTRODUCTION ....................................................................................................................... 1
1.1 Background of the Study..................................................................................................... 1
1.2 Theoretical Framework ....................................................................................................... 3
1.2.1 The Newtonian law of gravitation................................................................................ 3
1.2.2 Kepler‟s laws of planetary motion ............................................................................... 3
1.2.3 Three-body problem ..................................................................................................... 4
1.2.4 The circular restricted three-tody problem ................................................................... 4
1.2.5 Equilibrium points........................................................................................................ 4
1.2.6 Stability of equilibrium points ..................................................................................... 5
1.2.7 Periodic orbits .............................................................................................................. 7
1.2.8 Ellipse........................................................................................................................... 8
1.2.9 Eccentricity .................................................................................................................. 8
1.2.10 Special relativity......................................................................................................... 8
1.2.11 General relativity...................................................................................................... 11
1.2.12 Radiation pressure forces ......................................................................................... 12
1.2.13 Ellipsoid ................................................................................................................... 14
1.2.14 Oblateness ................................................................................................................ 15
1.2.15 Inertial and synodic coordinate systems................................................................... 15
1.2.16 Coriolis and centrifugal forces ................................................................................. 16
1.3 Statement of the Problem .................................................................................................. 17
x
1.4 Justification/Significance .................................................................................................. 19
1.5 Aim and Objectives of the Study ...................................................................................... 22
1.6 Methodology ..................................................................................................................... 22
1.7 Outline of the Thesis ......................................................................................................... 23
CHAPTER TWO ...................................................................................................................... 24
LITERATURE REVIEW......................................................................................................... 24
CHAPTER THREE .................................................................................................................. 31
OBLATENESS, RADIATION AND TRIAXIALITY OF THE PRIMARIES ................... 31
3.1 Introduction ....................................................................................................................... 31
3.2 Triangular Points with Radiation and Triaxial Primaries.................................................. 31
3.2.1 Equations of motion ................................................................................................... 31
3.2.2 Locations of the triangular points............................................................................... 35
3.2.3 Stability of L4,5 .......................................................................................................... 37
3.3 Triangular Points with Bigger Triaxial Primary and Smaller Oblate Primary .................. 48
3.3.1 Equations of motion ................................................................................................... 49
3.3.2 Locations of the triangular points............................................................................... 49
3.3.3 Stability of L4 ........................................................................................................... 52
3.4 Periodic Orbits Around Stable Triangular Points when the Bigger Primary is Triaxial ... 59
3.4.1 Elliptic orbits .............................................................................................................. 61
3.4.2 Orientation of the principal axes of the ellipse .......................................................... 61
3.4.3 Eccentricities of the ellipse ........................................................................................ 62
3.4.4 Semi-major and semi-minor axes............................................................................... 64
3.5 Collinear Equilibrium Points in the Relativistic Restricted Three-Body Problem with a
Smaller Triaxial Primary. ........................................................................................................ 65
3.5.1 Equations of motion ................................................................................................... 65
3.5.2 Locations of collinear points ...................................................................................... 66
3.5.3 Stability of collinear points ........................................................................................ 71
CHAPTER FOUR ..................................................................................................................... 74
PERTURBATIONS IN CORIOLIS AND CENTRIFUGAL FORCES, OBLATENESS
AND TRIAXIALITY OF THE PRIMARIES ........................................................................ 74
4.1 Introduction ....................................................................................................................... 74
4.2 Triangular Points with Perturbations in Coriolis and Centrifugal Forces with the Bigger
Primary as an Oblate Spheroid................................................................................................ 75
4.2.1 Equations of motion ................................................................................................... 75
4.2.2 Locations of the triangular points............................................................................... 76
xi
4.2.3 Stability of L4 ........................................................................................................... 80
4.3 Triangular Points with Perturbations in the Coriolis and Centrifugal Forces with a Triaxial
Bigger Primary ........................................................................................................................ 89
4.3.1 Equations of motion ................................................................................................... 89
4.3.2 Locations of the triangular points............................................................................... 90
4.3.3 Stability of L4 ........................................................................................................... 93
4.4 Collinear Equilibrium Points in the Relativistic R3BP with a Smaller Oblate Primary . 102
4.4.1 Equations of motion ................................................................................................. 102
4.4.2 Locations of collinear points .................................................................................... 104
4.4.3 Stability of collinear points ...................................................................................... 109
CHAPTER FIVE..................................................................................................................... 113
RESULTS AND DISCUSSIONS ........................................................................................... 113
5.1 Introduction ..................................................................................................................... 113
5.2 Results and Discussion of Section 3.2 ............................................................................ 113
5.3 Results and Discussion of Section 3.3 ............................................................................ 115
5.4 Results and Discussion of Section 3.4 ............................................................................ 117
5. 5 Results and Discussion of Section 3.5 ........................................................................... 118
5.5.1 Numerical results ..................................................................................................... 118
5.5.2 Results and discussion............................................................................................. 122
5.6 Results and Discussion of Section 4.2 ............................................................................ 123
5.7 Results and Discussion of Section 4.3 ............................................................................ 126
5.8 Results and Discussion of Section 4.4 ............................................................................ 128
5.8.1 Numerical results ..................................................................................................... 128
5.8.2 Results and discussion.............................................................................................. 131
CHAPTER SIX ....................................................................................................................... 133
SUMMARY, CONCLUSION AND RECOMMENDATIONS ........................................... 133
6.1 Summary ......................................................................................................................... 133
6.2 Conclusion ...................................................................................................................... 133
6.3 Recommendations ........................................................................................................... 134
REFERENCES ........................................................................................................................ 136
xii
LIST OF FIGURES
Figure 1.1: The location of the equilibrium points Li  i  1, 2,...,5
Figure 1.2: The rotating (synodic) coordinate system  0    with angular
velocity   n relative to the inertial (fixed) frame  0 X Y 
Figure 3.1: Reference parameter for collinear Lagrangian points
Figure 4.1: Reference parameter for collinear Lagrangian points
xiii
LIST OF TABLES
Table 3.1: Variation of discriminant  for case 1 of section 3.2
Table 3.2: Variation of discriminant  for case 2 of section 3.2
Table 3.3: Variation of discriminant  for case 1 of section 3.3
Table 3.4: Variation of discriminant  for case 2 of section 3.3
Table 4.1: Variation of discriminant  for case 1 of section 4.2
Table 4.2: Variation of discriminant  for case 2 of section 4.2
Table 4.3: Variation of discriminant  for case 1 of section 4.3
Table 4.4: Variation of discriminant  for case 2 of section 4.3
Table 5.1: Locations of the collinear points for the Sun-Earth system,
  0.000003003500, cd  10064.84, R  149597870.61km
Table 5.2: Locations of the collinear points for the Sun-Pluto system,
  0.00000006500, cd  63280.18, R  5900 106 km
Table 5.3: Locations of the collinear points for the Earth-Moon system,
  0.0121314293, cd  292624.8185, R  384000km
Table 5.4: Parameters of the systems
Table 5.5: Positions of the collinear equilibrium points
xiv
LIST OF SYMBOLS

Mass ratio
c
Critical value of the mass parameter
W
Potential force function
m1
Mass of the bigger primary
m2
Mass of the smaller primary
A1
Oblateness of the bigger primary
A2
Oblateness of the smaller primary
1
Distance from the third body to the bigger primary
2
Distance from the third body to the smaller primary
q1  1  1
Radiation coefficient of the bigger primary where 1 is the ratio of the
radiation pressure force to the gravitational force of the bigger primary
q2  1   2
Radiation coefficient of the smaller primary where  2 is the ratio of the
radiation pressure force to the gravitational force of the smaller primary
n
Mean motion
nd
Dimensionless mean motion
c
Speed of light
cd
Dimensionless speed of light
 i  i  1, 2 
Triaxiality coefficients of the bigger primary
 i  i  1, 2 
Triaxiality coefficients of the smaller primary
  1  1
Centrifugal force where 1 is the perturbation in the centrifugal force.
  1 2
Coriolis force where  2 is the perturbation in the Coriolis force
xv
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
The approximately circular motion of the planets around the sun and the small masses
of the asteroids and satellites compared to planetary masses suggested formulation of
the circular restricted three-body problem (CR3BP).
The restricted three-body problem (R3BP) represents a very wealthy treasure of
dynamical system, since the discovery of its non-integrability due to the pioneer
Poincare (1892-1899). This problem concerns with the motion of a test particle moving
under the gravitational effect of the two finite masses, called primaries, which move in
circular orbits around their center of mass on account of their mutual attraction and the
test mass does not influence the motion of the primaries. Although R3BP is insoluble,
or non-integrable, some families of particular solutions exist. Euler (1765) found a
collinear solution for the restricted three-body problem that assumes one of the three
bodies is a test mass. Soon after his solution was extended for a general three-body by
Lagrange, who also found an equilateral triangular solution in 1772. Now the solutions
for the restricted three-body problem are called Lagrange points. These solutions are
corresponding to the five singular points of Jacobian function. These families of
solution can be found by setting the derivatives of the potential to zero.
These five points are classified as three equilibrium collinear points Li i  1,2,3
(straight solution) or Eulerian points due to Euler and the other two equilibrium
Li i  4,5 represent equilateral triangle with the primaries.
1
On the line joining the primaries, the equivalent potential at the Eulerian points is a
maximum, and therefore these equilibrium points are unstable, the points are actually
saddle points.
The theory of the general relativity is currently the most successful gravitational theory
describing the nature of space and time, and well confirmed by observations. In
application to astrophysics, the general relativity enables one to analyse phenomena not
compatible within the framework of Newtonian concepts. General relativity made it
possible to calculate the binary pulsar motion (as a problem of relativistic celestial).
Binary pulsars observations confirmed the conclusion of general relativity about the
energy loss due to gravitational radiation. Krefetz (1976) computed the post-Netonian
deviations of the triangular Lagrange points from their classical positions in a fixed
frame of reference for the first time, but without explicitly stating the equations of
motion. After a decade, Brumberg (1972, 1991) studied the problem in more details and
collected most of the important results on relativistic celestial mechanics. The author
did not only obtain the equations of motion for general problem of three bodies but also
deduced the equations of motion for the restricted problem of three bodies. Contopolous
(1976) treated the relativistic R3BP in rotating coordinates. The author derived
Lagrangian of the system and the deviations of the triangular points as well. Maindl and
Dvorak (1994) derived the equation of motion for the relativistic R3BP using postNewtonian approximation of relativity. Abd EL-Salam and Abd EL-Bar (2011) derived
the equations of motion of the relativistic restricted three-body problem in the postNewtonian formalism.
2
1.2 Theoretical Framework
The fundamental laws of mechanics, as given by Newton in his Principia in 1687, may
be stated as:
First law: If there are no external forces, an object will maintain its state of motion, that
is it will stay at rest or continue rectilinear motion at constant velocity.
Second law: The rate of change of the momentum of an object is proportional to the
applied force F .
Third law: If a body A exerts a force F , on a body B, the body B will exert a force F ,
on body A.
1.2.1 The Newtonian law of gravitation
Every two particles in the universe attract each other with a force that is directly
proportional to the product of their masses and inversely proportional to the square of
the distance between them that is if the bodies A and B have masses M A and M B ,
respectively, and if their mutual distance is r , A will act on B with a force that is
directed towards A and has a magnitude G
M AM B
, where G is a constant of gravity,
r2
the value of which depends on the units chosen.
1.2.2 Kepler’s laws of planetary motion
i.
The planets move in ellipses with the Sun at one focus
ii.
A radius vector from the Sun to a planet sweeps out equal area in equal
times.
iii.
The square of the orbital period of a planet is proportional to the cube of its
semi-major axis.
3
1.2.3 Three-body problem
The three-body problem refers to three bodies of arbitrary masses which move under
their mutual gravitational attraction.
1.2.4 The circular restricted three-tody problem
If two of the three bodies move in circular and coplanar orbits around their common
barycenter, and additionally, the third body‟s mass is small compared to the other two
masses so that the third does not affect the movement of the other bodies, one speaks of
the circular restricted three-body problem. The two finite, massive bodies usually
denoted by m1 and m 2 , are called primaries and the third body (infinitesimal mass) is
called a test particle. The equations of motion of the classical restricted three-body
problem in the x, y, z  coordinate system with m1 and m 2 , and  1 and  2 , as the masses
and the distance to the bigger and smaller primaries respectively are:
x  2 y  Wx , y  2 x  Wy , z  Wz , where the force function is
x2  y2 1  

W


2
1
2
 i2  x  xi 2  y 2  z 2 , i  1,2,
x1   , x 2  1   , 0   
1
m2
,
.
2
m1  m 2
1.2.5 Equilibrium points
Although the circular restricted three-body problem is not integrable, we can find a
number of special solutions. This can be achieved by searching for points where the
particle has zero velocity and zero acceleration in the rotating frame. Such points are
called equilibrium points of the system.
4

L4
L1
L3
L2

m2  
m1  1  
L5
Figure 1.1: The location of the equilibrium points
Li  i  1, 2,...,5
1.2.6 Stability of equilibrium points
Five particular solutions of the motion of the infinitesimal body have been found. If the
infinitesimal body is displaced a very little from the exact points of the solutions and
given a small velocity it will either oscillate around these respective points at least for a
considerable time, or it will rapidly depart from them. In the first case the particular
solution from which the displacement is made is said to be stable; in the second case, it
is said to be unstable.
Let us have a dynamical system with s degree of freedom which is described by a set of
a differential equation as:
x  g  x  where x  R s and g   g1 , g2 ,...g s  is a vector function from R s into itself.
Now, let J be the Jacobian Matrix defined as:
5
 g1
 x
 1
 

J  x  Dg  x    
 

 g s
 x
 1
  



  
g1 
xs 

 

 
 

g s 
xs 
In the neighborhood an equilibrium location x  x0 , we refer to   J  x0   ,   R s as
the linearized system. According to the stability in Lyapunov sense solution x  x  t  is
stable if for any   0 there exists       0 such that if any solution y  y  t 
satisfy x  t0   y  t0    at the initial time t  t 0 then x  t   y  t    for t  t 0 , the
solution is said to be asymptotically stable
if for x  t0   y  t0    one has

lim x  t   y  t   0. as t 
When the solution coincides with an equilibrium point, its linear stability can be
inferred from the analysis of the eigenvalues of the Jacobian Matrix. Let i is the
'
eigenvector associated to the eigenvalue i  i  1, 2,...s  where, i s are the roots of the
characteristic equation det J  I   0 such that I is identity matrix of order s  s and
 i  R s then, the solution of the linearized system can written in the form:
s
  t    kii e t for some real or complex coefficient k i which are determined by the
i
i 1
initial conditions.
The stability properties for t  t 0 , of the linearized system may be stated simply:
(A) For complex roots of characteristic equation we have the following properties.
6
a. When the characteristic roots all have negative real parts, the equilibrium point
is asymptotically stable. This is true also when some of the roots are multiple.
b. When some or all the characteristic roots have positive real parts, the
equilibrium point is unstable. This is true also when some of the roots are
multiple.
(B) For pure imaginary roots the motion is oscillatory and the solution is stable
though it is not asymptotically stable. If there are multiple roots the solution
contains mixed (periodic and secular) terms and the equilibrium point is unstable.
(C) If the roots are real and all negative, the solution is stable, if any of the roots is
positive the point is unstable. These statements are also true for multiple roots.
Assuming for simplicity that s  2, the equilibrium points are characterized
according to the nature of the eigenvalues 1 and  2 as follows:
1. If 1 and  2 are real negative (positive) numbers, the equilibrium point is a
stable (unstable) node.
2. If 1 and  2 are real numbers with opposite signs, then the equilibrium point is
a sadde.
3. If 1 and  2 are complex numbers with negative (positive) real parts, then the
equilibrium is a stable (unstable) focus or spiral
4. If 1 and  2 are purely imaginary, then the equilibrium is a center or vortex.
1.2.7 Periodic orbits
A dynamical system in which the same configuration is repeated at regular interval of
time is said to exhibit a periodic motion. Same can be said of motions which are
repeated in a relative sense.
7
The periodic orbits have great significance generally in celestial mechanics and
especially in space dynamics.
For any particular solution of the restricted three-body problem, one can always find a
periodic solution. We can also get periodic orbits through linearized solution for the
motion of infinitesimal body around equilibrium point. Periodic orbits play a vital role
in separating the various classes of orbits and reduce the dimensionality of the problem
in phase space. In addition they can be used as reference orbits.
1.2.8 Ellipse
An ellipse is the locus of a point that moves such that the sum of its distances from two
fixed points called the foci is a constant.
1.2.9 Eccentricity
The orbital eccentricity of an astronomical body denoted by e , is the amount by which
its orbit deviates from a perfect circle. In other words the ratio of the distance between
the foci to length of the string is called the eccentricity e of the ellipse.
In other words the ratio of the distance between the foci to length of the string is called
the eccentricity e of the ellipse. From analytic geometry we know that when:
e  1, the orbit is an ellipse
e  1, the orbit is a parabola
e  1, the orbit is a hyperbola
e=0 , the orbit is a circle
1.2.10 Special relativity
In 1905, Albert Einstein introduced the special theory of relativity in his paper „On the
Electrodynamics of Moving Bodies‟. Special relativity, as it is usually called, postulates
two things: First, any physical law which is valid in one reference frame is also valid for
8
any frame moving uniformly relative to the first. A frame for which this holds is
referred to as an inertial frame. Second, the speed of light in vacuum is the same in all
inertial reference frames, regardless of how the light source may be moving.
The first postulate implies there is no preferred set space and time coordinates. For
instance, suppose you are sitting at rest in a car moving at constant speed. While
looking straight out a side window, everything appears to be moving so quickly!
Trees, building, and even people are flashing by faster than you can focus on them.
However, an observer outside of your vehicle would say that you are the one who
appears to be moving. In this case, how should we define the coordinates of you in your
car and the observer outside of your car? We could say that the outside observer was
simply mistaken, and that you were definitely not moving. Thus, his spatial coordinates
were changing while you remained stationary. However, the observer could adamantly
argue that you definitely were moving, and so it is your spatial coordinates that are
changing. Hence, there is no absolute coordinate system that could describe every event
in the universe for which all observers would agree and we see that each observer has
his own way to measure distances relative to the frame of reference he is in.
It is important to note that special relativity only holds for frames of reference moving
uniformly relative to the other, that is, constant velocities and no acceleration. We can
illustrate this with a simple example. Imagine a glass of water sitting on a table.
According to special relativity, there is no difference in that glass sitting on a table in
your kitchen and any other frame with uniform velocity, such as a car traveling at
constant speed. The glass of water in the car, assuming a smooth, straight ride with no
shaking, turning or bumps, will follow the same laws of physics as it does in your
kitchen. In this case, the water in each glass is undisturbed within the glass as time goes
9
on. However, if either reference frame underwent acceleration, special relativity would
no longer hold. For instance, if in your car, you were to suddenly stop, then the water in
your glass would likely spill out and you would be forced forward against your seat
belt.
Special relativity was not extended to include acceleration until Einstein published „The
Foundation of the General Theory of Relativity‟ in 1916. In special relativity, observers
in different inertial frames cannot agree on distances, and they certainly cannot agree on
forces depending on the distance between two objects. Such is the case with Newtonian
gravitation, as it describes gravity as an instantaneous force between two particles
dependent on their distance from one another. With this in mind, Einstein desired to
formulate gravity so that observers in any frame would agree on the definition,
regardless of how they were moving in relation to each other. Einstein accomplished
this by defining gravity as a curvature of spacetime rather than a force.
In physics, special relativity is the generally accepted and experimentally well
confirmed physical theory regarding the relationship between space and time. In
Einstein‟s original pedagogical treatment, it is based on two postulates:
1. That the laws of physics are invariant (i.e. identical) in all inertial systems (non
accelerating frames of reference).
2. That the speed of light in a vacuum is the same for all observers regardless of
the motion of the light source.
The theory is “special” in that it only applies in the special case where the curvature
of space-time due to gravity is negligible.
10
1.2.11 General relativity
There are three essential ideas underlying general relativity. The first is that space- time
may be described as a curved, four- dimensional mathematical structure called a
pseudo-Riemannian manifold. In brief, time and space together comprise a curved fourdimensional non-Euclidean geometry. Consequently, the practitioner of general
relativity must be familiar with the fundamental geometrical properties of curved spacetime. In particular, the laws of physics must be expressed in a form that is valid
independently of any coordinate system used to label points in space-time.
The second essential idea underlying general relativity is that at every space-time point
there exist locally inertial reference frames, corresponding to locally flat coordinates
carried by freely falling observers, in which the physics of general relativity is locally
indistinguishable from that of special relativity. This is Einstein‟s famous strong
equivalence principle and it makes general relativity an extension of special relativity to
a curved space-time. The third key idea is that mass curves space-time.
These three ideas have been demonstrated by contrasting general relativity with
Newtonian gravity. In the Newtonian view, gravity is a force accelerating particles
through Euclidean space, while time is absolute. From the viewpoint of general
relativity as a theory of curved space-time, there is no gravitational force. Rather, in the
absence of electromagnetic and other forces, particles follow the straightest possible
paths (geodesics) through a space-time curved by mass. Freely falling particles define
locally inertial reference frames. Time and space are not absolute but are combined into
the four- dimensional manifold called space-time.
In special relativity there exist global inertial frames. This is no longer true in the
presence of gravity. However, there are local inertial frames in general relativity, such
11
that within a suitably small space-time volume around an event, one may choose
coordinates corresponding to a nearly-flat space-time. Thus, the local properties of
special relativity carry over to general relativity.
General relativity generalizes special relativity and Newton‟s law of Universal
gravitation, providing a unified description of gravity as a geometric property of space
and time, or space time. General relativity predicts that the path of light is bent in the
gravitational field; light passing a massive body is deflected towards that body. This
effect has been confirmed by observing the light of stars or distant quasars being
deflected as it passes the Sun.
1.2.12 Radiation pressure forces
Cosmic radiation is the term that is used to describe the radiation that reaches us from
stars. While solar radiation is the term used to describe the radiation from our own Sun.
On the other hand the radiation pressure is the pressure exerted upon any surface
exposed to electromagnetic radiation.
So the dimensions of the actual body are very important when considering radiation
pressure coefficient. A large body can have, a large surface area, but if its diameter or
atomic diameters are not close to the incident radiation wavelength it will have a lower
radiation pressure coefficient, and thus a lower momentum will be imparted to it.
Generally if a beam of electromagnetic waves strikes a surface, waves will be reflected,
some will be absorbed and some keep going straight through, depending on the target‟s
properties and frequency. The radiation pressure coefficient is equal to the ratio of the
momentum acquired by the target to the electromagnetic momentum of the waves
before impact. For example when a beam of electromagnetic radiation is incident upon a
surface and complete transmission happened, the radiation pressure coefficient equal
12
zero. But its value equals one for total absorption and two when all incident waves are
reflected. Radiation pressure is dissipative force, the consequences of this force are loss
of energy and angular momentum of the satellite. A dissipative force leads to reduction
in the semi-major axis of the satellite.
As the solar radiation pressure force F p changes with the distance by the same law as
gravitational attraction force Fg and acts opposite to it, it is possible to consider that the
result of the action of this force will lead to reducing the effective mass of the massive
particle. Since the effect of reducing the mass of a particle depends upon the properties,
it is acceptable to speak about a reduced mass of the particle.
Thus the resultant force on the particle is
 Fp
F  Fg  F p  Fg 1 
 F
g

Where q  1 
Fp
Fg

  qFg


, is the mass reduction factor constant for a particle which depends on
the size and shape of the third body.
5.6 10 5
 where  is the particle
Chernikov (1970) has given the expression q  1 
d
radius and d its density, while  is a radiation-pressure efficiency factor (in the C.G.S.
system).
If the solar radiation flood fluctuations and a shadow effect of the planet are neglected,
then q is assumed to be constant. Depending upon the value of q , the reduced particle
mass is positive, negative or zero. In this thesis we consider the case when the
gravitation prevails, i.e. q  0 . We denote the radiation factors as qi i  1,2 for the
bigger
and
smaller
primaries
respectively,
13
and
they
are
given
by
 F pi
qi  1  
 Fg
 i

  1   i i  1,2 where  i is the ratio of radiation pressure force to the


gravitational force of the primary such that 0  1  qi 1  i  1, 2 
1.2.13 Ellipsoid
An ellipsoid is a closed type of quadratic surface that is a higher dimensional analogue
of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz Cartesian
x2 y 2 z 2
coordinate system is 2  2  2  1 , where a and b are the equatorial radii (along the
a b d
x and y axes) and d is the polar radius (along the z  axis ), all of which are fixed
positive real numbers determining the shape of the ellipsoid. If all three radii are equal,
the solid body is a sphere; if two radii are equal, the ellipsoid is a spheroid that is
a  b  d :Sphere
a  b  d :Oblate spheroid (disk-shaped)
a  b  d :Prolate spheroid (like a rugby ball)
a  b  d :Scalene ellipsoid (three unequal sides)
The points a,0,0 , 0, b,0 and  0,0, d  lie on the surface and the line segments from the
origin to these points are called the semi-principal axes. These correspond to the semimajor axis and semi-minor axis of the appropriate ellipses. Scalene ellipsoid are
frequently called “triaxial ellipsoid”, the implication being that all three axes need to be
specified to define the shape.
Any plane‟s cross section passing through the center of ellipsoid forms an ellipse on its
surface with the possible special case of a circle as the equator of revolving ellipsoid.
14
1.2.14 Oblateness
From a physical point of view, it is unreasonable to consider all objects as being points
masses with no physical dimensions. This is in conflict with the real cases for celestial
bodies. It is known that the effect of rotation causes deformation in the shape of the
objects at the equator as might be expected. In short the oblateness is the departure of
planets and celestial objects from spherical form because of the centrifugal force of the
rotation. For this reason most objects may be treated to a good approximation as oblate
spheroids (triaxial ellipsoid with two equal long axes and one short axis).
We denote Ai i  1,2 for the oblateness coefficients of the bigger and smaller
primaries
Ai 
respectively,
AE i2  APi 2
5R 2
such
i  1,2 , where
that 0  Ai  1
(McCuskey,
1963)
and
AE1 and AE 2 are the equatorial radii, AP1 and AP2 , the
polar radii of the bigger and smaller primaries respectively and R the distance between
the primaries.
1.2.15 Inertial and synodic coordinate systems
An inertial frame is a coordinate frame in which Newton‟s laws hold true. If we have an
absolute frame of the whole universe, at least all the frames moving at constant velocity
with respect to this absolute frame would be inertial. The concept of such an absolute
frame has, however, turned out to be rather problematic. Yet we can define that a frame
is inertial if experiments show that Newton‟s laws are valid in that frame.
The system of coordinates  ,  such that the plane rotates in the positive direction with
angular velocity equal to that of the common velocity of one primary with respect to the
other keeping the origin fixed, then that coordinate system is known as synodic system.
15
The primaries appear at rest in the synodic or rotaing system  ,  having the origin at
the centre of mass rotating along them and are placed on the   axis. The plane   is
the plane of motion of the primaries. The coordinates are sometimes called synodical. In
this system  ,  , the primaries m1 and m 2 are located at   ,0 , 1   ,0 respectively
and have zero velocity. The advantage of this system is that m1 and m 2 have fixed
positions, so that the equations of motion are time-independent and therefore it is the
easiest way to obtain the stationary solutions.
Figure 1.2: The rotating (synodic) coordinate system
(fixed) frame
 0    with angular velocity   n relative to the inertial
0 X Y 
1.2.16 Coriolis and centrifugal forces
In physics the Coriolis effect is a deflection of moving objects when they are viewed in
a rotating reference frame. In a reference frame with clockwise rotation the deflection is
to the left of the motion of the object, and for counter-clockwise rotation the deflection
is to the right. The Newton‟s laws of motion govern the motion of an object in a nonaccelerating inertial frame of reference. When Newton‟s laws are transformed to a
16
uniformly rotating frame of reference, the Coriolis and centrifugal forces appear. Both
the Coriolis and Centrifugal forces are proportional to the mass of the object. The
Coriolis force is proportional to the rotation rate and the centrifugal force is proportional
to its square. The Coriolis force acts in the direction perpendicular to the rotation axis.
The centrifugal force acts outwards in the radial direction and is proportional to the
distance of the body from the axis of the rotation frame. It is important to note that these
forces do not arise from any physical agency. They arise solely as a result of rotation of
the coordinate system. Thus, if the angular velocity is reduced to zero, both the Coriolis
and centrifugal forces are referred to as “fictitious or inertial” forces.
1.3 Statement of the Problem
The classical restricted three-body problem concerns the study of the motion of a
particle of infinitesimal mass (third body) in the gravitational field of two other massive
bodies, say m1 , m2 m1  m2  (conventionally called primaries). The classical restricted
three-body problem possesses five libration points (equilibrium points). The three
libration points L1 , L2 , L3 are collinear with the two primaries. The collinear points are
not stable in general for any value of the mass parameter  
m2
1
 . The two
m1  m2 2
libration points L4 , L5 define an equilateral triangle with primaries. They are called
triangular points. The triangular points are stable for the mass ratio 0     c and
1
unstable for c    , where  c is the critical mass parameter. This critical mass value
2
is used to study and analyze the size of the stability region. There exist also stable
periodic orbits of long and short periods at each of equilibrium point.
17
In order that the restricted three-body problem be realistic in real situations, the classical
problem was generalized by considering various aspects of perturbing forces such as the
shape of the bodies, radiation, Coriolis and centrifugal forces, and so on.
As in classical R3BP, in the relativistic restricted three-body problem also, the two
primaries have dominant masses and move around their center of mass; however the
third body is very small and its gravitational influence on the primaries is negligible.
The equations describing motion of the third body in the relativistic R3BP were
originally obtained by Brumberg (1972), who used a synodic (rotating) frame of
reference with the origin at the centre of mass and the primaries fixed.
The equilibrium points are the relativistic counterparts of the collinear L1 , L2 , and L3 ,
and triangular L4 and L5 points.
Linear stability of the relativistic collinear points was investigated by Ragos et al.
(2001) and Douskos and Perdios (2002), who showed that all these points were
unstable, which is consistent with the results obtained for the non-relativistic collinear
points.
In the work by Bhatnagar and Hallan (1998), linear stability of the relativistic triangular
L4 and L5 points was studied and it was demonstrated that these points were unstable
for the whole range 0   
1
, despite the well-known fact that the non-relativistic
2
L4 and L5 are stable for   0 , where 0  0.038521... is the Routh critical mass ratio.
The problem was later revisited by Douskos and Perdios (2002) and Ahmed et al.
18
(2006) who found that the relativistic triangular points are linearly stable in the range of
mass ratios 0    r where r  0 
17 69
(Douskos and Perdios, 2002), and
486cd2
r  0.03840 (Ahmed et al., 2006).
Hence due to the differences between the previous authors‟ results and also from our
knowledge no work has been done on the stability of equilibrium points in the
relativistic R3BP with perturbations, it raised a curiosity in our minds to generalize the
same model problem considering the problem “Investigation of stability of equilibrium
points in the relativistic relativistic restricted three-body problem with perturbations” to
comfirm or refute the previous authors‟ results. The problem is perturbed and
generalized in the sense that we considered:
1. Triaxiality of the bigger primary
2. Triaxiality of the smaller primary
3. Radiation pressures and triaxiality of the primaries
4. Triaxiality of the bigger primary and oblateness of the smaller primary
5. Small perturbations in the Coriolis and centrifugal forces with oblateness of the
bigger primary
6. Small perturbations in the Coriolis and centrifugal forces with triaxial bigger
primary
7. Oblateness of the smaller primary
1.4 Justification/Significance
The general theory of relativity was developed by Einstein a century ago. Since then, it
has become the standard theory of gravity, especially important to the field of
fundamental astrometry, astrophysics, cosmology and experimental gravitational
19
physics. Today, the application of general relativity is also essential for many practical
purposes involving astrometry, navigation, geodesy and synchronization.
The problem of computing relativistic effect in solar system is of increasing importance
for study of low perihelia and low semi major axis populations. The theory of general
relativity is currently the most successful gravitational theory of describing the nature of
space and time, and well confirmed by observations. Especially, it has passed “classical
tests”, such as the deflection of light, the perihelion shift of Mercury and the Shapiro
time delay, and also a systematic test using the remarkable binary pulsar “PSR”
1913+16 (Will, 1993). It is worthwhile to examine the three-body (or more generally,
N-body) problem in general relativity, N-body dynamics in the general relativity gravity
plays important roles in astrophysics. For instance, the formation of massive black holes
in star clusters is tackled mostly by Newton N-body simulations (Zwart et al., 2004). In
addition, future space astrometric missions such as Space Interferometry Mission (SIM)
and Galactic Astometric Instrument for Astrophysics (GAIA) require a general
relativistic modeling of the solar system within the accuracy of a microarc-second
(Klioner, 2003).
The Equations of motion of the relativistic R3BP by including various perturbing
parameters have been generalized.
The locations and stability of equilibrium points in the relativistic R3BP under the
following characterizations have been also:
20

When the primaries are triaxial and sources of radiation. It was found that the
relativistic terms, triaxiality and radiation factors affect the locations and reduce
the size of stability region.

When the bigger primary is oblate with small perturbations in the Coriolis and
centrifugal forces. It was found that the lacations are affected by the parameters
involved. It was also found that the relativistic terms, radiation factors and
centrifugal force all reduce the size of stability region and all have destabilizing
tendencies, while the stability behavior of the Coriolis force remains unchanged.

When the bigger primary is triaxial with small perturbations in the Coriolis and
centrifugal forces. It was found that the locations and stability region are all
affected by the parameters involved. All the parameters involved are found to
have destabilizing tendencies except the Coriolis force.
The collinear points L1 and L2 are the most interesting for space mission design due to
their fast instability. This instability has already been used in consuming small
expenditure of fuel presumably needed to keep a spacecraft there. Lagrange‟s triangular
points L4 and L5 for the Sun-Jupiter system are stable and indeed the Trojan asteroids
are located there.
21
1.5 Aim and Objectives of the Study
The aim of this research is to investigate the existence and linear stability of the
equilibrium points, together with periodic orbits around stable triangular points in the
relativistic R3BP with perturbations.
The aim has been achieved through the following objectives:
i. Generalization of the model of the relativistic restricted three-body problem.
ii. Determination of the locations of the equilibrium points
iii. Examination of the stability of equilibrium points
iv. Determination of long and short periodic orbits, their eccentricities and axes
around the triangular points.
1.6 Methodology
The method that is used to study the stability of equilibrium points is stated in the
following sequence:
i.
The subject of equilibrium points is approached by considering the problem
of finding location of the points where a particle (infinitesimal mass) could
be placed, with the appropriate velocity in the inertial frame, where it
remains stationary in the rotating frame.
ii.
In order to investigate the stability of equilibrium points we obtain the
variational equations of motion using the classical approach.
iii.
Then the characteristic equation and its roots are obtained. Using
Lyapunov‟s criteria, the nature of stability of the point under consideration is
determined.
iv.
Subsequently, the value of the critical mass parameter is computed.
v.
Finally, the region of stability is established.
22
1.7 Outline of the Thesis
This thesis is divided into six chapters. The first and second are the introductory chapter
and the literature review respectively. The main body of the work is divided into two
chapters (chapter three and four): Oblateness, radiation and triaxiality of the primaries
are studied in chapter three while small perturbations in the Coriolis and centrifugal
forces together with oblateness and triaxiality are studied in chapter four. Finally, the
discussions of the results are given in chapter five and summary, conclusion,
recommendations are given in chapter six.
23
CHAPTER TWO
LITERATURE REVIEW
The general three-body problem is the problem of motion of three celestial bodies under
their mutual gravitational attraction. The restricted three-body is a simplified form of
the general three-body problem, in which one of the bodies is of infinitesimal mass, and
therefore does not influence the motion of the remaining two massive bodies called the
primaries (Bruno, 1994;Valtonen & Karttunen, 2006).
The circular restricted three-body problem (CR3BP) possesses five stationary solutions
called Lagrangian points. Three are collinear with the primaries and the other two are in
equilateral triangular configuration with the primaries. The three collinear points L1 ,2 ,3
are unstable, while the triangular points L4 ,5 are stable for the mass ratio

m2
 0.03852..., m1  m2 being the masses of the primaries (Szebehely, 1967a).
m1  m2
(Wintner,1941;Contopoulos, 2002) have shown that the stability of the triangular
equilibrium points is due to the existence of the Coriolis terms in the equations of
motion when these equations are written in rotating coordinate system. In the classical
problem, the effects of the perturbations have been ignored. Perturbations can well arise
from the causes such as from the lack of sphericity, or the triaxiality, oblateness,and
radiation forces of the bodies, variation of masses, the atmospheric drag, the solar wind,
Poynting Robertson effect and the action of the other bodies. The most striking example
are perturbations due to oblateness in the solar system is the orbit of the fifth satellite of
Jupiter, Amalthea. This planet is very oblate and the satellite‟s orbit is very small that its
line of apsides advances about 9000 in one year (Moulton, 1914). Such oblatenessdriven effects are competing disturbing effects for qualitatively similar general
24
relativistic effects (Iorio, 2009; Iorio et al., 2013;Iorio, 2006;Renzetti, 2012b).The
Kirkwood gaps in the ring of the asteroid‟s orbits lying between the orbits of the Mars
and Jupiter are examples of the perturbations produced by Jupiter on an asteroid. This
enables many researchers to study the restricted problem by taking into account the
effects of small perturbations in the Coriolis and the centrifugal forces ,radiation,
oblateness and triaxiality of the bodies (Szebehely, 1967a; Schuerman, 1972; SubbRao
and Sharma ,1975; Bhatnagar and Hallan ,1978; Bhatnagar and Hallan, 1979;
Schuerman, 1980; AbdulRaheem and Singh ,2006; Oberti
and Vienne, 2003;
AbdulRaheem and Singh, 2008, Singh, 2011a,b; Singh and Begha, 2011;Singh, 2013;
Abouelmagd, 2013). Szebehely (1967b) investigated the stability of triangular points by
keeping the centrifugal force constant and found that the Coriolis force is a stabilizing
force.
The bodies in the R3BP are strictly spherical in shape, but in nature, celestial bodies are
not perfect spheres. They are either oblate or triaxial. The lack of sphericity, triaxiality
or oblateness of the celestial bodies causes large perturbation in a two body orbit. It
must be noted here that the asphericity of the primary has a number of applications in
several fields of astronomical sciences (e.g testing alternative gravities) and also in
fundamental physics (Renzetti, 2012a; Iorio, 2009). The asphericity issue inspired
several researchers (Subbarao and Sharma, 1975; Elipe and Ferrer, 1985; El-Shaboury
and El-Tantawy, 1993) to include non-spherity of the bodies in their studies of the
R3BP. When one or both primaries are triaxial bodies this problem was discussed by ElShaboury et al. (1991), Khanna and Bhatnagar (1999), Hallan et al. (2001), Sharma et
al. (2001a, 2001b), Singh (2013). Khanna and Bhatnagar (1999) have discussed the
stationary solutions of the planar restricted three-body problem when the smaller
primary is a triaxial rigid body with one of the axes as the axis of symmetry and its
25
equatorial plane coinciding with the plane of motion. The bigger primary is taken as an
oblate spheroid and its equatorial plane coinciding with the plane of motion. They have
shown that there exist five libration points, two triangular and three collinear. The
collinear points are unstable, while the triangular points are stable for the mass
parameter 0     crit (the critical mass parameter) and the triangular points have long
or short periodic elliptical orbits in the same range of  . Sharma et al. (2001b) have
discussed the existence of the libration points in the restricted three-body problem when
both the primaries are triaxial rigid bodies and shown that there exist five libration
points, two triangular and three collinear. Sharma and Rao (1975) investigated
numerically the collinear libration points, by taking the oblateness of the primaries in
consideration for 19 systems. They found that in some of the systems the shifts are
significant. These equilibria are shown to be unstable in general, though the existence of
conditional, infinitesimal (linearized) periodic orbits around them can be established, in
the usual way. They also showed that the eccentricity and synodic period of these orbits
are functions of oblateness. Numerical study, in this connection, with the above
systems, revealed that the orbits around the libration point which is farthest from the
primary whose oblateness effect is included, exhibit a different trend from those around
the other two points. Singh and Umar (2014) investigated the motion of a test particle in
the vicinity of a binary made of a triaxial primary and spherical companion moving
along elliptic orbits about their common barycenter in the neighborhood of the collinear
libration points. Their positions and stability are found to be affected by triaxiality of
the bigger primary and by the semi-major axis and the eccentricity of the binary‟s orbits
as well. They obtained the analytical results and applied it to binary neutron stars
consisting of a bigger triaxial primary and a spherical companion.
26
Abouelmagd and El-Shaboury (2012) studied the existence of libration points and their
linear stability when three participating bodies are axisymmetric and the primaries are
radiating, they found that the collinear points remain unstable, and the triangular points
are stable for region 0     c ; the range of stability for these points decreases. They
studied periodic orbits around the triangular points and found that these orbits are
elliptical; the frequencies of long and short orbits of the periodic motion are affected by
terms which involve the parameters that characterize the oblateness and radiation
repulsive forces; they deduced that the period of long periodic orbits adjusts with the
change in its frequency while the period of short periodic orbits will decrease.
The theory of general relativity is currently the most successful gravitational theory
describing the nature of space and time, and well confirmed by observations
(Will,2014) . Regarding the three–body relativistic effects we may also cite: The
geodesic precession of the orbit of two-body system which is about a third mass in
general relativity (Renzetti, 2012a) and for the post-Newtonian tidal effects (Iorio,
2014).
In 1967, Krefetz computed the post-Newtonain deviations of the triangular Lagrangian
points from their classical positions in a fixed frame of reference for the first time, but
without explicitly stating the equations of motion. After a decade, Contopoulos (1976)
treated the relativistic R3BP in rotating coordinates. He derived the Lagrangian of the
system and the deviations of the triangular points as well.
Then, Brumberg (1972, 1991) studied the relativistic n -body problem of three bodies in
more detail and collected most of the important results on relativistic celestial
mechanics. He did not only obtain the equations of motion for the general problem of
three bodies but also deduced the equations of motion for the restricted problem of three
27
bodies. Maindl and Dvorak (1994) derived the equation of motion for the relativistic
R3BP using the post-Newtonian approximation of relativity. They applied this model to
the computation of the advance of Mercury‟s perihelion in the solar system and found
that they are compatible with published data.
Bhatnagar and Hallan (1998) studied the existence and linear stability of the triangular
points L4,5 in the relativistic R3BP, and found that L4,5 are always unstable in the whole
range 0   
1
in contrast to the classical R3BP where they are stable for   0 ,
2
where  is the mass ratio and  0  0.03852... is the Routh‟s value.
In the beginning of the 21st century, Ragos et al. (2001) investigated numerically the
linear stability of the collinear libration points L1,2,3 in the relativistic R3BP for several
solar system cases, and found that the points L1,2,3 are unstable. Douskos and Perdios
(2002) examined the stability of the triangular points in the relativistic R3BP and
contrary to the result of Bhatnagar and Hallan (1998), they obtained a region of linear
17  69  2
1
stability in the parameter space as 0    0 
486cd 2
, where  0  0.03852... . is
Routh‟s value. They also determined the positions of the collinear points and showed
that they are always unstable.
Later, Wanex (2003) studied the chaotic amplification in the relativistic R3BP and
noticed that the difference between Newtonian and post-Newtonian trajectories for the
restricted three-body problem is greater for chaotic trajectories than it is for trajectories
that are not chaotic. He also discussed the possibility of using this chaotic amplification
effect as a novel test of general relativity.
28
Ahmed et al. (2006) investigated also the stability of the triangular points in the
relativistic R3BP. In contrast to the previous result of Bhatnagar and Hallan (1998),
they obtained a region of linear stability as 0    0.03840 , where 0 is the Routh‟s
value.
There after, Abd El-Salam and Abd El-Bar(2011) derived the equations of motion of the
relativistic three-body in the post-Newtonian formalism.
Yamada and Asada (2012) discussed the post-Newtonian effects on Lagrange‟s
equilateral triangular solution for the three-body problem .For three finite masses,it is
found that a triangular configuration satisfies the post-Newtonian equation of motion in
general relativity ,if and only if it has the relativistic corrections to each side length
.This post-Newtonian configuration for three finite masses is not always equilateral and
it recovers previous results for the restricted-problem when one mass goes to zero. They
also found that for the same masses and angular velocity,the post-Newtonian triangular
is always smaller than the Newtonian one. Yamada and Asada (2011) investigated
collinear solutions to general relativistic three-body problem. They showed that the
equation determining the ratio among the three masses, which has been obtained as a
seventh-order polynomial has at most three positive roots, which apparently provide
three cases of the distance ratio. It is found however, that even for such cases, there
exists one physically reasonable root and only one, because the remaining two positive
roots do not satisfy the slow motion assumption in the post-Newtonian approximation
are discarded. This means that especially for the restricted three-body problem, exactly
three positions of a third body are true even at the post-Newtonian order. Abd El-Bar
and Abd El-Salam (2012) investigated the relativistic effects on the equilibrium point of
the relativistic R3BP. They obtained approximate locations of collinear and triangular
29
points. Abd El-Bar and Abd El-Salam (2013) computed the locations of collinear points
in the photogravitational relativistic R3BP. Series forms of these locations are obtained
as new results. Lastly, Abd El-Salam and Abd El-Bar (2014) studied the
photogravitational Restricted Three-Body Problem within the framework of the postNewtonian approximation. The mass of the primaries are assumed to change under the
effect of continuous radiation process. The locations of the triangular points are
computed. Series forms of the locations are obtained as new analytic results.
From literature above and also to our present knowledge, no work has been done on the
stability of equilibrium points on the relativistic R3BP with the perturbations, hence, it
raised a curiosity in our minds to study the effects of various perturbations on the above
mentioned problem .
30
CHAPTER THREE
OBLATENESS, RADIATION AND TRIAXIALITY OF THE
PRIMARIES
3.1 Introduction
In this chapter, the equations of motion taking into consideration the oblateness,
triaxiality and radiation of the primaries are presented. The locations are obtained and
the study of the stability of the equilibrium points is carried out. The periodic orbits
around the stable triangular points when the bigger primary is triaxial are also
examined.
3.2 Triangular Points with Radiation and Triaxial Primaries
In this section, the locations and stability of triangular points when the primaries are
triaxial and radiating sources are studied.
3.2.1 Equations of motion
The equations of motion of the infinitesimal mass in the relativistic restricted threebody problem in a barycentric synodic coordinate system  ,  with origin at the centre
of mass of the primaries are given by Brumberg (1972) and Bhatnagar and Hallan
(1998) and Ragos et al. (2001):
  2n 
W d  W
 
 dt  



(3.1)
  2n 
W d  W
 
 dt  



where,
31
 m m  1 1
1
W  n 2  2   2    1  2   2   2   2  2n(  )  n 2  2   2
2
 1  2  c  8




 2   2  2n(  )  n 2  2   2 
2




n 2 a 2 2
2M
2


2

3
2
 m1 m 2 



 1  2 
 m12 m 22   m1 m 2  
7  1
1 
 2  2 

na 4  n   


M  
2    1  2 
 1 2 
 m 2 m1 

m 2  2m1 m1  2m 2 
a
 3  3   n 2 a 2  







2
M

2
M


1
2
1
2


2 
 1
(3.2)
  constant of gravitation

n  mean motion  1 

mm
3M 
1  1 22
2
2c a 
3M
  M

3

a2
(3.3)
m1 , m2  masses of the primaries
1 ,  2  distances of the infinitesimal mass from the primaries
c  velocity of light
M  m1  m2
a  distance between
the primaries.
The above mentioned dimensional quantities are transformed to dimensionless ones
when the units of mass, length and time are chosen such that M  1 , a  1 and   1
respectively.
Then, if we denote by  , 0   
1
, the mass of the less massive primary, the mass of
2
the other primary is equal to 1   and their coordinates are 1   ,0 and   ,0
respectively.
In dimensionless system, the equations of motion given by (3.1) remain the same, but
the values of W and n given in equation (3.2) and equation (3.3) become respectively
32
W
1 2
1 

1  3 1
1

  2 

 2  1   (1   ) ( 2   2 )   2   2  2(  )
2
1
 2 cd  2  3
8




  2  2

2

3 1  



2  1
2
 2
1  (1   ) 2  2 
    2  2(  )   2   2  
 2 
2   12
2 



7  1
1  2

  (1   ) 4    

2    1  2  2


  1   
1
3  2 1  3 
 3  3    
 



2 1
2  2 
 2   1  2
 1
(3.4)
and
nd  1 
3  1     
c
1
,
 , cd 
2 
2cd 
3
  m1  m2 

a
(Ragos et al, 2001)
(3.5)
where,
12  (   ) 2   2
(3.6)
  (    1)  
2
2
2
2
The triaxiality and radiation factors of the bigger and smaller primaries are included
with the help of the parameters  i ,  i ,  i (i  1, 2) , respectively.
The pertinent equations of motion of an infinitesimal mass in the relativistic R3BP in a
barycentric synodic coordinate system  ,  and dimensionless variables can be written
as:
  2nd 
W d  W 
 

 dt   
(3.7)
  2nd  
W d  W 
 

 dt   
with
33
q (1   ) q2  q1 (1   )  2 1   2 
1 3
3

W  1   2 1   2    2 1   2   ( 2   2 )  1


2 2
2
1
2
2 13


2
3q1 (1   )  2 1   2  2 q2   2 1   2  3q2   2 1   2 
1


 2
5
3
5
2 1
22
22
cd

2
1 2 2
3  q (1   ) q2    2 2
2
2

    2(  )  ( 2   2 )   1

     2(   )  (   )
8
2  1
2 


 3 1
 2 2
  2 1  3  (1   )  (   )

 



1  q 2 (1   ) 2 q22  2 
7   1 1   2  q  q (1   )   1
  1 2
 2   q1q2 (1   )   4         2 3  1 3   
2
1
2 
2   1  2  2  1
 2   1 2


q2   2q1 (1   ) q1 (1   )  2q2   

 
2 1
2 2
 
(3.8)
nd  1 
3
3
3
1
 21   2    21   2   2 1   (1   ) 
4
4
2cd  3

(3.9)
12  (   ) 2   2
(3.10)
  (    1)  
2
2
where 0   
2
2
1
is the ratio of the mass of the smaller primary to the total mass of the
2
primaries, 1 and  2 are distances of the infinitesimal mass from the bigger and smaller
primary, respectively; nd is the mean motion of the primaries; cd is the dimensionless
speed of light.
h2  f 2
b2  f 2
h2  f 2
b2  f 2
1 
, 2 
,  1 
,  2 
, (McCuskey, 1963) and
5R 2
5R 2
5R 2
5R 2
 i  1,  i  1 (i  1,2) characterize the triaxiality of the bigger and smaller primary
with h, b , f as lengths of the semi-axes of the bigger primary and h, b, f  as those of
the smaller primary. The radiation factor qi (i  1,2) is given by Fpi  Fgi (1  qi ) such that
0  (1  qi )  1 (Radzievskii, 1950), where Fgi and Fpi are respectively the gravitational
and radiation pressure.
Here as Katour et al. (2014), the parameters  i , 1 (i  1,2) are not included in the
relativistic part of W since the magnitude of these terms is very small due to cd 2 .
34
3.2.2 Locations of the triangular points
The libration points are obtained from equation (3.7) after putting         0.
These points are the solutions of the equations
W
W
with     0.
0


This is equivalent to.
q2  (  1   ) 
3q (1   ) (   )(2 1   2 )
3

 3  1   1    2   2     1
3

2
2
2 15


15q1 (1   ) (   )( 2   1 ) 2 3q2  (  1   )(2 1   2 ) 15q2  (  1   )( 2   1 ) 2



2 17
2  25
2  27


q1 (1   )(   )
3
1

q  (  1   ) 
1 
3 2
  (1   )  1
2
2
2  q1 (1   )(   )
3 1 
 2
   (   )  (   ) 

2 
3
cd 
3  2
2
1
 23




 q (1   ) q2  
q12 (1   ) 2 (   ) q22  2 (  1   )
1 
7  1
3  1


 q1q2  (1   )    
 
4
4
2 
1
2

 1
 2  1  2 
7  (   ) (  1   )  3 2  q2  (   ) q1 (1   )(  1   )  (   ) (  1   )
  



  
 3
2 
13
 23
15
 25
1  23
 2 
 1  2

 q2   2q1 (1   )  (   )   q1 (1   )  2q2   (  1   )   0
2 13


2  23
and
(3.11)
F  0,
with
2
 q (1   ) q  
3q (1   )  3
3
 15q1 (1   )  2   1 
F  1  1 3  2 3   3  1   1    2   2   1 5


2


2
1

1
2 
2
1
2 17
2


2
 q1 (1   ) q2  
3q2   3
1    (1   )  1 2
 15q2   2   1 
2


 5   2  2 1  
 2  3 1 


  (   )  3 
7
2  2
2 2
cd  
3
2 

 2
 1

 q (1   ) q   q 2 (1   ) 2 q22  2
3
1   q  q (1   ) 
7  1
 ( 2   2 )  1 3  2 3   1 4
 4  q1q2  (1   )     3  3    2 3  1 3 
2
2 
1
2
2 

 1
 2  1  2   1
 q   2q1 (1   )    q1 (1   )  2q2    
3  q  q (1   )   1
1
 2  25  1 5    3 
 2
 
3
2  1
 2   1  2 1  2
2 13
2  23


35
The triangular points are the solutions of the system (3.11) with   0 . Since
and in the case
1
 1
cd 2
1
 0 and in the absence of triaxiality and radiation (i.e.
cd 2
 i   i  i  0, i  1, 2 ), one can obtain 1  2  1 , and we assume in the relativistic
R3BP with radiation pressure and triaxiality that 1  1  x and 2  1  y where
x, y  1 may be depending upon the radiation, triaxiality, and relativistic terms.
Substituting these values in equation (3.10), solving them for  , , and ignoring terms of
second and higher powers of x and y , the following solutions are obtained as
  x  y
 3
1  2
,
2
  

2

x y
.
3 
(3.12)
Now substituting the values 1 , 2 ,  , in equations (3.11) with   0 and for
simplicity, putting qi  1  1  qi   1   i and neglecting second and higher terms in
x, y,
1
,  i ,  i,  i (i=1,2) and their products, the following system is obtained as
cd 2
1 1     2   57 2  69 1 
3
1
 3
2 




 1        1   2   x    1 1    
y
2
2 
2
2
16

2
 2
57 2  69 1 

45
3
1  9 27  2 9 3 


  1   2  
   2 1   2   2  


0
16
16
4
cd  16
16
8 
(3.13)
3 1     3    1   x  3  3  y  1    
1

2
1
  2 
3
21
 1  3 2     1   2 
8
8
3
3
1  21
 1  3 2     2 1   2   2   1     0
2
2
cd  8

Solving these equations for x and y , the following solutions are obtained as
36
x
  2  3  1 11
y
1    5  3    2  
8cd

2
8cd
3

8
2
3

 2  1   1 


2 1   
 1
 2
 1   1 

 
1     2

 1

1 3
11
11
  1   
 1  2   1   2

8
8
 2 2 
 2 
(3.14)
Thus , the coordinates of the triangular points  ,   denoted by L4 and L5
,respectively,are

3
 1 3
1  2 
5  1 1 
 
 
  2   
1 
  1
1  
2 
2  4cd   8 2 
 2 8 
 8 2(1   ) 
 
7
1

   2   2  1 
3
 2(1   ) 8 
 
 19 1  
3
1
2  23 1 
2

5

6


6








1 

1


 
 2 
2  12 cd 2
3  8 2 
 8 2  
 15
2  19
 
 
   
  1   
  2
3  8 2(1   ) 
8
2(1


)




 2
    2  1  
 9


(3.15)
3.2.3 Stability of L4,5
3.2.3.1 The Variational equations
In

order
to

study
the
motion
near
any
of
the
equilibrium
points
L  0 , 0 , 0 , 0  La, b,0,0, we may write   a   ,  b   ,  ,   1.
The function W may be expanded in a Taylor series around L giving

1
0 2
0
0

W  W 0  W0  W0   W0  W0  
W0  2  W0  2  W0 2  W
    2W   W
2!
   W 0   ...
W 0  W 0   W 0 





(3.16)
The superscript 0 means that these derivatives are calculated at the equilibrium point
0 ,0 ,0,0 .
Then, the following equations are obtained
37
W  W0   W0   W0  W0 
 A  B  C  D
(3.17)
where, A  W0 , B  W0 , C  W0 , D  W0
W  W0  W0  W0  W0   W0 
(3.18)
d  W 

  W0  W0  W0   W0   F  B2   C2  D2 

dt   
(3.19)
where,
F  W0 , B2  W0 , C2  W0 , D2  W0
W  W0   W0   W0   W0  E  B1  C1  D1
(3.20)
where,
E  W0 , B1  W0 , C1  W0 , D1  W0
W  W0  W0   W0   W0   W0 
(3.21)
d  W

dt  
(3.22)

  W0   W0   W0   W0   A3  B3   C3  D3 

where,
0
A3  W0 , B3  W0 , C3  W0 , D3  W

Since the nature of the linear stability about the point L5 will be similar to that about L4 ,
it will be sufficient to consider here the stability only near L4 .
Let (a,b) be the coordinates of the triangular point L4
In equations (3.7), we set   a   ,  b   , ( ,   1) .
First, the terms of the R.H.S of the above equations are computed, neglecting second
and higher order terms, using equation (3.17), the following expression is obtained as
38
 W 


 A  B  C  D



  a  , b  
(3.23)
where,
3 15 2  19  8 
3  31 2    8 
3
1
2 
A  1 
1 
2
 2  19  19  
4  2cd 2
16
16


3 15 2  49  26 
16(1   )
 1 
3  31 2  63  24 
16(1   )
 3 
 1  2 ,
 2

1
2
 2  (3  1)1  
3  89 2  47   8 
3  37  2  9  8 

3 3
2 
B
1 
2
1  2  1  2  
4
16
16
 3cd 

3  89 2  131  50 
16(1   )
C
3
1  2  ,
2cd 2
D
6  5  5 2
.
2cd 2
 1 
3  37  2  65  36 
16(1   )
 2 
3
3
(1   )1 
(2   ) 2 ,
6
6
Similarly, using equation (3.20), the following equation is written as
 W 
 E  B1  C1  D1





 a  , b 
(3.24)
where,
3  89 2  47   8 
3  37  2  9  8 

3 3
2 
E
1 
2
1  2  1  2  
4
16
16
 3 cd 

3  89 2  131  50 
16(1   )
 1 
3  37  2  65  36 
16(1   )
39
 2 
3
3
(1   )1 
(2   ) 2 ,
6
6
3 15 2  29  8 
3 15 2  7   8 
9
7
2 
B1  1  2  2  3  3  
1 
2
4  6cd
16
16


C1 
3 15 2    22 
16(1   )
 1 
3 15  23
1
 3 
 2  (1  3 )1    1  2 ,
16
2
 2

1
4     2  ,
2 
2cd
D1  
3 1  2 
2cd 2
.
Using equation (3.19), the following equation is written as
d  W 
 F  B2   C2  D2 


dt    a  ,  b  
(3.25)
where,
F
3
1  2  ,
2cd 2
B2  
C2 
1
4    2 ,
2 
2cd
1
17  2  2 2  ,
2 
4cd
D2  
3
1  2  .
4cd 2
Similarly , using equation (3.22) the following equation is written as
d  W 


 A3  B3   C3  D3 

dt     a  , b 
40
(3.26)
where,
A3 
1
 6  5  5 2  ,
2cd 2
B3  
3
1  2  ,
2cd 2
C3  
3
1  2  ,
4cd 2
D3 
3  5  2  2 2 
4cd 2
.
Thus the variational equations of motion corresponding to equations (3.7), on utilizing
equation (3.9), can be written as:
p1  p2   p3  p4   p5  p6   0,
(3.27)
q1  q2   q3  q4   q5  q6   0.
where,
p1  1  C2 , p2  D2 , p3  F  C




 3
3
3  1



p4   B2  2 1   2 1   2   2 1   2 
1   (1   )    D  ,
2 
4
2cd  3



 4


p5   A, P6   B


 3
3
3  1

q1  C3 , q2  1  D3 , q3  2 1   2 1   2   2 1   2 
1   (1   )  
2 
4
2cd  3

 4
 C1  A3 , q4  B3  D1 ,
q5   E , q6   B1
41
3.2.3.2 Characteristic equation
The system (3.27) can be written in matrix form as follows:
Let
1  
 2  
1  
(3.28)
 2  
Substitute equation (3.28) in equation (3.27), to obtain
p1 2  p2 2  p3 2  p4  2  p51  p6 1  0,
(3.29)
q1 2  q2 2  q3 2  q4 2  q51  q6 1  0.
Solving for derivatives for 1 ,  2 , 1 , 2 and noting that p3  q4  0 , the following
system is obtained as
 2  
2 
p2 q3
pq pq
pq pq
p4 q2
2 
 2  2 5 5 2 1  2 6 6 2 1
p1q2  q1 p2
p1q2  q1 p2
p1q2  q1 p2
p1q2  q1 p2
p1q3
q p pq
q p pq
q1 p4
2 
 2  1 5 1 5 1  1 6 1 6 1
p1q2  q1 p2
p1q2  q1 p2
p1q2  q1 p2
p1q2  q1 p2
1   2
(3.30)
1   2
The system in equation (3.30) corresponds to matrix form:
p2 q3

 2   p1q2  q1 p2
   
p1q3
 2   
1   p1q2  q1 p2
 
1
 1 

0


p4 q2
p1q2  q1 p2
p2 q5  p5 q2
p1q2  q1 p2
q1 p4
p1q2  q1 p2
q1 p5  p1q5
p1q2  q1 p2
0
0
1
0
The characteristic equation of the system is given as:
42
p2 q6  p6 q2 
p1q2  q1 p2   2 

q1 p6  p1q6    2 
p1q2  q1 p2  1 
 
  1 
0

0

(3.31)
p4 q2
p1q2  q1 p2
p2 q5  p5 q2
p1q2  q1 p2
p2 q6  p6 q2
p1q2  q1 p2
q1 p4

p1q2  q1 p2
q1 p5  p1q5
p1q2  q1 p2
q1 p6  p1q6
0
p1q2  q1 p2
1
0
0
0
0
1
0
0
p2 q3

p1q2  q1 p2


p1q3
p1q2  q1 p2
(3.32)
It is important to note that
q1  p2 , q3   p4 , q5  p6
Hence the characteristic equation can be written in the form
( p1q2  p2 q1 ) 4  ( p1q6  p5q2  p6 q1  p2 q5  p4 q3 ) 2  p5q6  p6q5  0
(3.33)
Substituting the values of pi , qi , i  1, 2...,6 and neglecting second and higher powers of
small quantities, the characteristic equation of the variational equations of motion
corresponding to equation (3.7) can be expressed as:
4  b2  d  0
(3.34)
where,

9 
3
3

b  1  2   3 1  (2  3) 2  3 1   3    2
2
2

 cd 
2
3
2
27  (1   ) 9  65  77   24  12  9  89  99  10 
d


1
4
8cd 2
16

9  37  2  47   10 
16
2 
9  89  79 
9  37   27 
 1 
 2
16
16
3
3
  (1   2 )   2 (1   2 )
2
2
43
1
0
cd 2
For
and
when
the
primaries
are
non-luminous
and
spherical
(i.e. 1   2   1   2   1   2  0), equation (3.34) reduces to its well-known
classical restricted problem form (see e.g. Szebehely, 1967a):
4  2 
27
 (1   )  0.
4
The discriminant of equation (3.34) is

54 4 108 3 
801
333
801
333
693  2
  2    27  61  6 2 
1 
2 
 1 
 2 

2
cd
cd
4
4
4
4
2 cd 2 


891
447
771
219
585 
57
63
  27  61  6 2 
1 
2 
 1 
 2  2    1   1   2
4
4
4
4
2 cd 
2
2

18
 6 1  3 2  2
cd
(3.35)
The roots of equation (3.34) are
2 
b 
2
(3.36)
where,

9 
3
3

b  1  2   3 1  (2  3) 2  3 1   3    2
2
2

 cd 
From (3.35), the followings are obtained

d  216 3 324 2
801
333
801
333
693 
 2   2   2  27  61  6 2 
1 
2 
 1 
 2 

2 
d
cd
cd
4
4
4
4
2
c
d 


891
447
771
219
585 
 1
  27  61  6 2 
1 
2 
 1 
 2 
 0     0,  .
2 
4
4
4
4
2 cd 
 2

(3.37)
44

d 2
648
648
801
333
801
333
693 
  2  2  2   2  27  61  6 2 
1 
2 
 1 
 2  2   0
2
d
cd
cd
4
4
4
4
3cd 

 1
    0, 
 2
(3.38)
d
 1
is monotonic increasing in  0, 
d
 2
This implies that
But
 d 
891
447
771
219
585
 1
1 
2 
 1 
 2  2  0     0, 

  27  61  6 2 
4
4
4
4
2cd
 2
 d    0
(3.39)
 d 
45
57
15
57

 1    1   2   1   2
2
2
2
2
 d   
(3.40)
2
and
    0  1 
    1  
2
57
63
18
 1   2  6 1  3 2  2  0
2
2
cd
23 525
57
525
57
3
207

1   2 
 1   2  (1   2 ) 
0
4 16
16
16
16
2
4 cd 2
In order to study the monotonicity of  , two cases are considered:
 d 
Case 1: 
 0
 d    1
2
Fot this case, the table of variation of  is given in table below
45
(3.41)
Table 3.1: Variation of 

0
1/2
d 2
d2

d
d


    0
   1 2
 1
From the above table it can be seen that  is monotonic decreasing in  0,  Since
 2
.
   0 and     1
are of opposite signs, and  is monotone and continuous and by
2
 1
intermediate value property there is one value of  say. c in  0,  for which   0 .
 2
 d 
Case 2: 
0

 d    1
2
 d 
Since from equation (3.38), 
 is monotonic increasing in
 d 
 d 
 1
 0,  and 
 0
 2
 d    0
 d 
 d 
 1
0
 0,
and 
  0 , this implies that there exists    0,  such that 

 2
 d     12
 d    0
hence
d
1
d

 0     0 , 
 0     0,  0  and
d
2
d

Hence the following table of variation of  is shown below
46
Table 3.2: Variation of 

0
0
12

d
d
 d  d   1/2

 d  d    0

    0
    1
2
    
Since
   0  0 and     1  0 ,
0
it can be concluded that
2
    
0
 0 . Since  is
continuous and monotone in  0,  0  and by the intermediate value property there is one
value of  say. c in  0,  0  for which   0 . Hence c  c  c . Solving the
equation   0 using equation (3.35), the critical value of the mass parameter is obtained
as
1 1
17 69 1  5
59 
1  19
85 
15
59 
69 
  
1   
2   
 1


2
2 18
486cd
2  6 9 69 
2  18 9 69 
2  6 9 69 
1  19
85 
2
  
 2 
(1   2 )

2  18 9 69 
27 69
c  
(3.42)
17 69 1  5
59 
1  19
85 
15
59 
  
1   
2   
 1


2
486cd
2  6 9 69 
2  18 9 69 
2  6 9 69 
1  19
85 
2
  
 2 
(1   2 )

2  18 9 69 
27 69
c  0 
(3.43)
47
where  0  0.03852... is the Routh‟s value.
The following three regions of the values of  are considered separately.
When 0    c ,   0, the values of 2 given by equation (3.36) are
i.
negative and therefore all the four characteristic roots are distinct pure
imaginary numbers. Hence, the triangular points are stable.
1
When  c    ,   0 , the real parts of the characteristic roots are positive.
2
ii.
Therefore, the triangular points are unstable.
iii.
When    c ,   0 , the values of 2 given by equation (3.36) are the same.
Therefore the solutions contain secular terms.This induces instability of the
triangular points.
Hence, the stability region is
0    0  p
(3.44)
with
p
17 69
2
1 5
59 
1  19
85 
1 5
59 

(1   2 )   
 1   
 2   
 1
2
486c
2  6 9 69 
2  18 9 69 
2  6 9 69 
27 69
1  19
85 
  
 2
2  18 9 69 
3.3 Triangular Points with Bigger Triaxial Primary and Smaller Oblate
Primary
In this section the same methodology as in section 3.2 is followed.
The triaxiality of the bigger primary and oblateness of the smaller primary are
introduced by means of parameters  i (i  1,2) and A2 respectively with  i  1i  1,2,
A2  1, where  1 
( AE 2  AP 2 )
h2  f 2
b2  f 2
,


.
A

, (McCuskey,1963). Here
2
2
5R 2
5R 2
5R 2
 1 ,  2 characterize the triaxiality of the bigger primary with h, b, f as lengths of its
48
semi-axes, AE and AP are the equatorial and polar radii of the smaller primary, and R
is the distance between the primaries.
3.3.1 Equations of motion
Neglecting second and higher power of  i (i  1,2) , A2 and also their products, the
equations of motion can be written as:
  2nd 
W d  W 
 

 dt   
(3.45)
  2nd  
W d  W 
 

 dt   
where W is the potential-like function of the relativistic R3BP. As Katour et al. (2014),
we do not include the parameters  i (i  1,2), A2 in the relativistic part of W since the
magnitude of these terms is so small due to cd 2 .
Hence,
A  1 
1 3
3
1   

W  1  A2   2 1   2   ( 2   2 ) 
  1  22  
 2 1   2 
2 2
2
1
 2  2  2  2 13


3 1    2
1
 2   1   2
5
2 1
cd
 3 1
1 2
 2
2
2

  2 1  3  (1   )  (   )  8     2   






2
3  1     2
1  (1   ) 2  2 
( 2   2 )  
     2  2     ( 2   2 )  
 2
2  1
2 
2  12
2 





7  1
1   2   1   
1
3  2 1  3   


  (1   )  4         3  3    


  ,
2   1  2  2  1
 2   1  2
2 1
22  



(3.46)
and nd the perturbed mean motion of the primaries is given by
nd  1 
3
3
3
1
 21   2   A2  2 1   (1   ) 
4
4
2cd  3

(3.47)
3.3.2 Locations of the triangular points
The libration points are obtained from equation (3.45) after putting         0.
49
These points are the solutions of the equations
W
W
with     0.
0


That is

(1   )(   )

3
1

3(1   ) (   )(2 1   2 )
 (  1   ) 3   (  1   )  
3 
 A2  
   3 1   2   
3
5
2
2 
2
2 
2 15
 

15(1   ) (   )(2 1   2 ) 2 1 
3 2
  (1   )  1
2
2
2  (1   )(   )
 2  3 1 
   (   )  (   ) 
7
2 1
cd 
3  2
2
13



 7  1 1  7  (   )
 (  1   )   1    
(1   ) 2 (   )  2 (  1   )

3




  (1   )        



3
4
4
2
2 
1
2
13
 2  1  2  2 
  1

(  1   )  3 2   (   ) (1   )(  1   )  (   ) (  1   )  3  2  (   )



  
 3
5
 23
 25
1  23
2 13
 2  1
 1  2

1  3  (  1   )   0
2  23


and
F  0,
(3.48)
with
2
 1    3 
  
3  3(1   )  3
 15(1   )  2   1 
F  1  3  3   A2 1  5    3 1   2  
  2 1  
1
2  2  2  
2 
15  2 2
2 17



1
cd 2
1  
   (1   )  1 2
2

  (   )  3 
 3 1 
3
2
 2
 
 1
 3 2
   (1   ) 2  2 
2 1 

(



)

 4

 3

 23   14
2 
 2
 1
7  1
 3  2   1  3  
1    1   3   1  
1
1

  (1   )     3  3    3  3    2  5  5   3 


 2  2  1
 2  1  2 1 23
2 13
2  23 

 2  1  2   1
The triangular points are the solutions of equations (3.48) with  0. Since
in the case
1
 1 and
cd 2
1
 0 and in the absence of triaxiality and oblateness factors
cd 2
i.e. 1   2  A2  0 , one can obtain 1  2  1 ; we assume in the relativistic R3BP
that 1  1  x and  2  1 y where, x, y  1. may be depending upon the relativistic
50
and triaxiality factors. Substituting these values in the equations (3.6), solving them for
 , and ignoring terms of second and higher powers of x and y , the solutions are
obtained as
  x  y
1  2
,
2
 3 x y
.

2
3 

  
(3.49)
Substituting the values of 1 ,  2 ,  , from the above equations (3.48) with   0 , and
neglecting the terms of second and higher powers in x, y,
x y
,
, 1,  2 , A2 and their
cd 2 cd 2
products, the following system is obtained as
 57 2  69 1    45     1   9  27  2  9 3   0
3
3
3
1    x  y  A2 (1   ) 
 1 2 2 

2
2
4
16
16
cd  16
16
8 
21   
3
3
21
3 1    x  3 y  A2 (1   )   1  3 2     1   2  
2
8
8
8cd 2
2
 0
(3.50)
Solving these equations for x and y , the solutions are
x
y
  2  3  1
11
 A2  ( 2   1 )
2
8cd
2
8
1    5  3 
8cd
2
(3.51)
 1 3
 1


 1   
 1  2
 2 2 
 2 
Thus, the coordinates of the triangular points  ,  denoted by L4 and L5 respectively
are,
51

1 1 
 1 3
1  2 
5  1
 A2   
  2
1 
1  
2 
2  4cd  2
 8 2 
 2 8 

 19 1  
1
2  23 1 
3 
 3

2

5

6


6









A2  .


1 




 1 
 2
2
3  8 2 
 8 2    6


 2  12 cd

(3.52)
  
3.3.3 Stability of L4
In this section the same methodology as in section (3.2.3) is followed.
Let a, b  be the coordinates of the triangular point L4
Setting   a   ,  b   , ( ,   1) , in the equations (3.45)
The terms of their R.H.S. are computed, neglecting second and higher order terms, the
following equations are obtained
 W 


 A  B  C  D



  a  , b 
(3.53)
where,
3 15 2  19  8
3  31 2    8
3
1
2   3  24  
A  1 
1 
2,
 2 19  19    8  A2 
4  2cd 2
16
16

3 89 2  47   8
3  37  2  9  8

3 3
2   3(7  26 ) 
B
1 
2,
1  2  1  2   
 A2 
4
8
16
16

 3cd  
C
3
1  2  ,
2cd 2
D
6  5  5 2
.
2cd 2
Similarly,
52
 W 
 E  B1  C1  D1





 a  , b 
(3.54)
where,
3  89 2  47   8 

3 3
2   3(7  26  ) 
E
1
1  2  1  2   
 A2 
4
3
c
8
16



d  


3  37  2  9  8 

16
2,
3 15 2  29  8
3 15 2  7   8 
9
7
33
2 
B1  1 
1 
2,
 2  3  3   8 A2 
4  6cd 2
16
16

C1 
1
 4     2  ,
2cd 2
D1  
3 1  2 
2cd 2
.
d  W 
 F  B2   C2  D2 
 
dt    a  , b  
where,
F
3
1  2  ,
2cd 2
B2 
1
4     2  ,
2 
2cd
C2 
1
17  2  2 2  ,
2 
4cd
D2  
3
1  2  .
4cd 2
53
(3.55)
d  W 


 A3  B3   C3  D3 
dt     a  , b 
(3.56)
where,
A3 
1
6  5  5 2  ,
2 
2cd
B3  
3
1  2  ,
2cd 2
C3  
3
1  2  ,
4cd 2
D3 
3  5  2  2 2 
4cd 2
.
Following as in section (3.2), the variational equations of motion corresponding to
equations (3.45), on making use of equation (3.47), can be obtained as
p1  p2   p3  p4   p5  p6   0,
q   q   q   q   q   q   0.
1
2
3
4
5
(3.57)
6
where


 3
3
3  1



p1  1  C2 , p2  D2 , p3  F  C , p4   B2  2 1   2 1   2   A2 
1


(1


)

D
,



2
4
2cd  3



 4


p5   A, p6   B
 3
3
3  1

q1  C3 , q2  1  D3 , q3  2 1   2 1   2   A2 
1   (1   )    A3  C1 , q4  B3  D1 ,
2 
4
2cd  3

 4
q5   E , q6   B1
54
Then, the corresponding characteristic equation is
( p1q2  p2 q1 ) 4  ( p1q6  p5q2  p3q4  p6q1  p2q5  p4q3 ) 2  p5q6  p6q5  0
(3.58)
Substituting the values of pi , qi , i  1, 2,...,6 in equation (3.58), the characteristic
equation (3.58) after normalizing becomes
4  b2  d  0
(3.59)
where,

9  
3
3
b  1  2    3   A2  3 1   2  3  2 ,
2
2
 cd  
2
3
9  89 2  99  10 
27  (1   ) 9  65  77   24  12  117
d


 (1   ) A2 
1
4
8cd 2
4
16

9  37  2  47   10 
16
When
 2.
1
 0 and in the absence of the triaxiality and oblateness (i.e. 1   2  A2  0),
cd 2
equation (3.59) reduces to its well-known classical restricted problem form (see e.g.
Szebehely, 1967a):
4  2 
27  (1   )
0
4
The discriminant of equation (3.59) is

54 4 108 3 
801
333
693  2
  2    27  117 A2 
1 
2 

2
cd
cd
4
4
2 cd 2 


891
447
585 
18 57
63
  27  123 A2 
1 
2 
  1  3 A2  2   1   2
2 
4
4
2 cd 
cd
2
2

(3.60)
Its roots are
2 
b 
2
(3.61)
where,
55

9  
3
3
b  1  2    3   A2  3 1   2  3  2 ,
2
2
 cd  
From (3.60), we have

d  216 3 324 2
801
333
693 
 2   2   2  27  117 A2 
1 
2 

d
cd
cd
4
4
2 cd 2 


891
447
585 
 1
1 
2 
 0     0,  .
 27  123 A2 
2 
4
4
2 cd 
 2

(3.62)

d 2
648
648
801
333
693 
 1
  2  2  2   2  27  117 A2 
1 
 2  2   0     0, 
2
d
cd
cd
4
4
2cd 
 2

(3.63)
d
is monotonic increasing in
d
This implies that
 1
 0, 
 2
But
 d 
711
219
585
 1
1 
 2  2  0     0, 

  27  123 A2 
4
4
2cd
 2
 d    0
(3.64)
 d 
45
57
162

 1  6 A2   1   2  2
2
2
cd
 d   
(3.65)
2
   0  1  3 A2 
    1  
2
57
63
18
1   2  2  0
2
2
cd
23 207 117
525
57


A2 
1   2  0
2
4 4 cd
4
16
16
In order to study the monotonicity of  , two cases are considered:
 d 
Case 1: 
 0
 d    1
2
For this case ,the table of variation is given below:
Table 3.3: Variation of 
56
(3.66)

0
1/2
d 2
d2

d
d


   0
   1 2
 1
From the above table it can be seen that  is monotonic decreasing in  0,  .
 2
Since
    0
and     1 are of opposite signs, and  is monotone and continuous
2
and by the intermediate value property, there is one value of  say. c in the interval
 1
 0,  for which   0 .
 2
 d 
Case 2: 
0

 d    1
2
 d 
Since from equation (3.63), 
 is monotonic increasing in
 d 
 d 
 1
 0,  and 
 0
 2
 d    0
 d 
 1
0
and 
 1  0 , this implies that there exists    0,  such that
 2
 d   
2
hence
 d 
 0,


 d    0
 d 
d
 0 1
 0     0,  0  and 
  0      ,  , hence the following table of
2
d

 d 
variation is shown below.
57
Table 3.4: Variation of 

0
0
12

d
d
 d  d   1/ 2

 d  d    0

    0
  1
2
    
0
Since    0  0 and     1  0 , it can be concluded from the above table that
2
    
0
 0 ,hence since     0 and
decreasing and continuous in
 0, 
0
    
0
are of opposite signs, and  is monotone
 and by the intermediate value property, there is
one value of  say. c in  0,  0  for which   0 . Hence c  c  c .
Solving the equation   0 , using equation (3.60), the critical value of the mass
parameter is obtained as
1 1
17 69 1 
13 
15
59  1  19
85 
69 
 1 
A2   
  
2


2
2 18
486cd
9
2  6 9 69  2  18 9 69 
69 
c  
(3.67)
c   0 
17 69 1 
13 
15
59 
1  19
85 
 1 
A2   
1   
2


2
486cd
9
2  6 9 69 
2  18 9 69 
69 
where  0  0.03852... is the Routh‟s value.
The following three regions of the values of  are considered separately.
58
When 0    c ,   0, the values of 2 given by equation (3.61) are
i.
negative and therefore all the four characteristic roots are distinct pure
imaginary numbers. Hence, the triangular points are stable.
ii.
1
When  c    ,   0 , the real parts of the characteristic roots are positive.
2
Therefore, the triangular points are unstable.
iii.
When    c ,   0 , the values of 2 given by equation (3.61) are the same.
Therefore the solutions contain the secular terms.This induces instability of
the triangular points.
Hence, the stability region is
0    0 
17 69 1 
13 
15
59 
1  19
85 
 1 
A2   
1   
2


2
486cd
9
2  6 9 69 
2  18 9 69 
69 
(3.68)
3.4 Periodic Orbits Around Stable Triangular Points when the Bigger
Primary is Triaxial
The study of periodic orbits is considered as one of the most important aspect in the
restricted three-body problem; and is a starting point for attacking the problem of
classifying solutions. Furthermore, if one has a particular solution for the restricted
problem, then he can always get a periodic solution. The importance of studying these
orbits lies in their significant appearing in nature, they provide us exciting informations
about orbital resonances and spin orbits and they can be used as reference orbits to
approximate quasi-periodic trajectories.
In this section the results of the problem (3.3) when smaller primary is spherical (i.e.
A2  0 ) are used
Hence the triangular Lagrangian points L4,5  0 , 0  are given from equation (3.52) as:
0 
1  2 
5
1 
2 
4cd 2
0  
 1 1 
 1 3
  2
 
1  
8
2

2

8




 19 1  
3
1
2  23 1 
5  6  6 2    

1 
1   
  2 
2 
2  12 cd
3 59
 8 2   
 8 2  
(3.69)
In section 3.2, it is stated that the triangular point are linearly stable in the range
0    c and the characteristic equation has four pure imaginary roots. Thus, the
motion in this region is bounded and made up of two harmonic motion with frequencies
s1 and s 2 given by:
  C1 cos s1t  D1 sin s1t  C2 cos s2t  D2 sin s2t
(3.70)
  C1 cos s1t  D1 sin s1t  C2 cos s2t  D2 sin s2t
with
Ci  i  2nsi Di  W0 Ci 
 i  1, 2 
 i  1, 2 
Di  i  2nsi Ci  W0 Di 
and
i 
s i2  W0
4n 2 s i2  W0 
2

(3.71)
i  1,2
1
s  W0
2
i
(Szebehely,1967a)
where s1 and s 2 are the frequencies with respect to long and short periodic orbits
respectively, the terms with coefficients C1 , D1 , C1 , D1 are the long periodic terms while
the coefficients C 2 , D2 , C 2 , D2 are the short periodic terms.
The four roots of the characteristic equation (3.59) when A2  0 are written as:
1,2  i s1
and
(3.72)
3,4  i s2
From equation (3.61) when A2  0 we get
60
s12 
27  (1   )  45 891
801 2 
333 2 
 45 423



 1   

  2
4
16
16
16
 8

 8 16

1
 2 108 4  216 3  693 2  585 
8cd
s22  1 
27  (1   )  69 891
801 2 
333 2 
 81 471
 

 1    

  2
4
16
16
 8 16

 8 16

1
 2 108 4  216 3  693 2  585  72 
8cd
(3.73)
3.4.1 Elliptic orbits
The Taylor expansion of the function of the potential W around the triangular points
L4,5 can be written in the following form:
1
1
W  W 0  W0  2  W0   W0  2  0  3 ,  3 
2
2
(3.74)
where the terms containing the third and higher powers of  and  are neglected and
W0 
1 2
3 
1
 11 11
    3       2   1   6 2    1  2

2
2 
8
8 8
1
 2  3 4  6 3  2 2  5  3 ,
8cd
(3.75)
3 15 2  19  8
3  31 2    8
3
1
2 
W  1 
1 
 2 , (3.76)
 2 19  19  
4  2cd 2
16
16

0
3 15 2  29  8
3 15 2  7   8 
9
7
2 
W  1 
1 
 2 , (3.77)
 2  3  3  
4  6cd 2
16
16

0
3 89 2  47   8
3  37  2  9  8

3 3
2 
W 
1 
 2 , (3.78)
1  2  1  2  
4
16
16
 3cd 
0
Equation (3.74) represents a quadratic form and indicates that the periodic orbits around
the triangular points are ellipse.
3.4.2 Orientation of the principal axes of the ellipse
With the transformation
61
    cos 
 
     sin 
tan 2 
sin    
  , the directions of the principal axes are given by:
cos    
2W0
W0  W0
[Abouelmagd and El-Shaboury (2012)]
(3.79)
Substituting (3.76), (3.77) and 3.78) in (3.79), we obtain
 45 3  7  2  40  16 
10

2
tan 2   3 1  2  
1

2

1

3


3







1
3cd 2
6



 69 3  25 2  24  16  

 2  ,
6

 
(3.80)
3.4.3 Eccentricities of the ellipse
The Taylor expansion of the function of potential W around the triangular points L4,5 is
given by equation (3.74), but the Jacobian constant C  2W implies that
1
1
 11 11

C    2    3      3 2   1  1    6 2   2 
3 4  6 3  2 2  5  3
2 
4
4cd
4 4

 3 
 15 2  19  8 
 31 2    8   2
1 
2
  1 
2

19


19


3


3
  16  1  16   2 
2 
 4  2cd 



 
 3 3

 89 2  47   8 
 37  2  9  8  
2 

1  2  1  2   3 
1  3 
  2  
8
8



 
 3cd 
 2
 15 2  29  8 
 15 2  9  8   2
7
 9 
2 
  1 

2

3


3


3


3


 
 1 
 2   .
2
16
8


 
 
 4  6cd
(3.81)
The characteristic equation of the associated matrix of equation (3.81) is



3
3
 2  3  9 1   3    2 
2
4c

801


 6        274  1       458  891
16
16
2

d


333 2 
153
 45 423
 

   2  2  1     0
16
8cd
 8 16

2
(3.82)
Its roots are
62
2

1

1  3 
2 
9 1     87 297
267 2 
111 2 
3
 27 93
 

 1      
   2  2  12  19  19 2 
4
16
16 
8cd
 8 16

 8 16
9 1     15 297
267 2 
111 2 
51
 15 141
 

 1   

  2  2   2   
4
16
16 
8cd
 8 16

 8 16
(3.83)
The eccentricities of the ellipse are given by (Szebehely, 1967a)

ei  1   i2

1
2
, i 
2s i
s 
2
i
i  1,2
(3.84)
where  is one of the roots of equation (3.82). For i  1,2 and making use of equation
(3.73) and   1 , we obtain
 5
87
671
267

5
95
391
12  3 1     9 2 1         
 2  282 3 
 4 1     
2
2
4
4
2
2
4
4



2
132 3 
111 4 
1
  2 
47   518 2  1032 3  741 4  270 5  90 6 
2 
2
2cd

2
1 9 1    243 1     9 1377
4713 2 3807 3 7209 4 
  

 

 
 
 1
4
8
32
64
32
16
64
 32

2685 2 1755 3 2997 4 
1
 27 1269
 

 
 
  2 
72  414  20679 2
2 
64
32
16
64
32cd
 32

2
2
2
41364 3  29187  4  10206 5  3402 6  .
(3.85)
Hence,
3
9
671 2
267 4 
391 2
 5 87
 5 95
e1  1   1      2 1        
  141 3 
  1      

2
2
8
4
8
4 8

 4 8
66 3 
111 4 
1
  2 
47   518 2  1032 3  741 4  270 5  90 6 
2 
4
4cd

(3.86)
63
2
3  3 1    81 1     3 459
1571 2 1269 3 2403 4 
e2 
1


 

 
 
 1

2 
4
16
16
8
32
 16 32

2
895 2 585 3 999 4 
1
 9 423
 

 
 
  2 
 24  138  6893 2
16
8
32
64cd 2
 16 32


13788 3  9729 4  3402 5  1134 6  .
3.4.4 Semi-major and semi-minor axes
The lengths of the semi-major axis, ai and the semi-minor axis, bi of the long and short
period are obtained from
1

 2 2
ai    02  02 
i 

and
(3.87)
1
bi   i2 02  02  2
respectively, with  0 and 0 as initial conditions (Szebehely, 1967a).
To obtain the semi-major axes of the orbit of the long period and that of the short
period, the equations (3.69) and (3.85) are used in equation (3.87), when i  1,
respectively. Thus, the followings are obtained
a1 
 221
5 
1 7  2   493
33
25
3 
9  2 25
3 

 7   2 





2




1 


1

 2
2
2  10 10   120
10
24  20  
10 24  20  2 
 120


1 
42
517  1020   156  2  324  3  108 4  .
2 
60cd 


13  25
629 2 729 3 729 4   4 4135
3
21573 2 16767 3
 
 
  


 

1   
2  13
52
26
52
13
104
52
  13 104
a2 


 323 1884
8019 4 
3
58563 2 37017 3 31347 4 
 1  



 
 
  2
52
13
13
104
52
104


 52

1
402  2312  124901 2  251640 3  180495 4  65610 5  21870 6  
2 
104cd

(3.88)
64
Similarly, the semi-minor axes of the orbits of the long and short periods are obtained as
b1 
3  
39 3 43 4
277
1 477 2 1151 3
2
5
6   10


 

1   4      18  6     
2  2
3
2
3
8
6
  3 24



1991 4 557 5
261
1 1039 2 215 3 943 4 241 5

 
  89 6   1   3 


 
 
 

6
2
24
3
24
2
6
2


111 6 
1
  2 
52  209  204 2  2714 3  6296 4  6822 5  3954 6
2 
3
12cd


1440 7  360 8 
and
b2 
13  17
31 2 585 3 1015 4 729 5 243 6   161 1889
3507 2
 
 
 
 
 



1   
4  52
208
104
208
52
52
104
  208 416

 205 1701
11337 3 22533 4 45117 5 7209 6 
4
8055 2 4443 3
 
 
 
 1  



 

104
104
208
104
13 416
208

 208 416

987 4 8565 5 2997 6 
1
 
 
  2 
 312  422  21353 2  114036 3  247209 4
52
208
104
416cd 2


 263250 5  151254 6  54432 7  13608 8 
(3.89)
3.5 Collinear Equilibrium Points in the Relativistic Restricted Three-Body
Problem with a Smaller Triaxial Primary.
In this section, the analytical and numerical locations of the collinear points are obtained
and also the study of their stability when the smaller primary is triaxial is carried out.
3.5.1 Equations of motion
The triaxiality factors of the smaller primary are introduced with the help of the
parameters  i  1, i  1,2 where  1 
h2  f 2
b2  f 2
,


(McCuskey, 1963) with
2
5R 2
5R 2
h, b, f as lengths of its semi-axes and R is the dimensional distance between the
primaries.
Ignoring second and higher powers of  i and neglecting also their product, the
equations of motion are taken as:
65
W d  W 

  2n 
 
 dt   
(3.90)
W d  W 

  2n 
 
 dt   
where W is the potential like function of the relativistic R3BP. As Katour et al. (2014),
the parameters  i (i  1, 2) are not included in the relativistic part of W since the
magnitude of these terms is so small due to cd 2 where cd is the dimensionless speed of
light.
Hence,
1 3
1  

3 2

W  1   2 1   2   ( 2   2 ) 
  3  2 1   2  
 2   1 
2 2
1  2 2  2
2  25


1
cd 2
 3 1
1 2
 2
2
2
2
2

  2 1  3  (1   )  (   )  8     2     (   )







2
3  1     2
1  (1   ) 2  2 
 
     2  2     ( 2   2 )  
 2
2  1
2 
2  12
2 




(3.91)
7   1 1   2   1     1 3  2 1  3  

  (1   )  4         3  3   


  ,
2   1  2  2  1
 2   1  2
2 1
2  2  

and nd is the perturbed mean motion of the primaries and is given by
nd  1 
3
3
1
 21   2   2 1   (1   ) 
4
2cd  3

(3.92)
3.5.2 Locations of collinear points
Equilibrium points are those points at which no resultant force acts on the third
infinitesimal body. Therefore, if it is placed at any of these points with zero velocity, it
will stay there. In fact all derivatives of the coordinates with respect to the time are zero
at these points. Therefore, the equilibrium points are solutions of equations
66
W  0 and W  0
(3.93)
W and W may be written as
W   


1
cd 2
(1   )(   )
13

3 (  1   )(2 1   2 ) 15 (  1   )( 2   1 ) 2
 (  1   ) 
3 

3






2
 1
 23
2 
2  25
2  27


3 2
 (  1   )   1    
  (1   )  1
2
2
2  (1   )(   )

 
   (   )  (   ) 
  3
 3 1 
3
3  2
2
1
 23
2 



  1
(1   ) 2 (   )
14

 2 (  1   )
 7  1 1  7  (   ) (  1   ) 
  (1   )        


4
2
13
23
 2  1 2  2 

3   (   ) (1   )(  1   )  (   ) (  1   )  3  2  (   ) 1  3  (  1   ) 
 2 





 3
2  15
 25
1 23
2 13
2  23
 1  2

and
W  F ,
with
2
 1    
3  3  3
 15  2   1 
F  1  3  3    3 1   2   5   2  2 1  
1
2  
2  2  2
2  27


1
cd 2
1    3 2
   (1   )  1 2
   (1   ) 2  2 
2
2 1 

3
1


(



)

3


(



)

 4
 3

 3


 
3
 23  2
 23   14
2 
 2
 
 1
 1
 7  1
 3  2   1  3   .
1    1   3   1  
1
1
  (1   )     3  3    3  3    2  5  5   3 


3
 2  2  1
 2  1  2 1  2
2 13
2  23 
 2  1  2   1
In order to find the collinear points, we put   0 in equation (3.93). Their abscissae are
the roots of the equation
67
g     

1
cd 2
(1   )(   )

3
1

3 (  1   )(2 1   2 )
 (  1   ) 
3 
  3 1   2   
3
2
2 
2  25


  (1   )  1 3 3 2  (1   )(   )  (  1   )   1    

 
  3
    
 3 1 
3
2 
13
 23
2 

 2

  1
(3.94)
 7  1 1  7  (   ) (  1   ) 
(1   ) (   )  (  1   )


  (1   )        


4
4
1
2
13
 23
 2  1  2  2 

2

2
  1       3  2  (   ) 1  3  (  1   ) 
 3 

  0
1 23
1  2
2 13
2  23

with 1     ,  2    1  
To locate the collinear points on the   axis, the orbital plane is divided into three parts:
  1 , 1    2 and  2   with respect to the primaries where 1  
and
 2  1 
m1  1  
m2  
1
L1
(a)
1 ,0   ,0
 2 ,0  1  ,0
C
m1  1  
L2
 ,0
2
m2  
(b)
1 ,0   ,0
L3
1  3
 ,0
C
 2 ,0  1  ,0
m2  
m1  1  
(c)
 ,0
1 ,0   ,0
68
C
 2 ,0  1  ,0
Figure 3.1: Reference parameter for collinear Lagrangian points
Case1. Position of L1    2  (see Fig. 3.1 (a))
Let   2  1;   1  1  1    1  1  1; since the distance between the primaries
is unity, i.e.  2  1  1  1   and  2  1   then
  1  1  ; 1  1  1; 2  1 with  i  0 i  1,2
(3.95)
Now substituting equation (3.95) in equation (3.94), the following equation is obtained
110   6  3  19   9  2cd 2  13  3cd 2 2   2  6cd 2 1  18   24cd 2 1  2 2  12cd 2 2
3 cd 2 2  16  8cd 2  1   3  6 cd 2 1  2 cd 2  17   18cd 2 2  9 cd 2 2  3 3  18 cd 2 1
6 cd 2  12cd 2  12  36cd 2 1  16   7  2  24cd 2 1  18 cd 2 1  6cd 2  16  6 cd 2  9  3 3
12cd  2  9 cd  2    16  6 cd  1  14  3cd  2  6cd  1 3 cd  2  6 cd  3  
2
2
5
1
2
2
2
2
2
2
3
(3.96)
4
1
  3 cd 2 2  12 2  15  6 cd 2  3 3  6 cd 2 1  13   5  9 cd 2 2  2 cd 2

18 cd 2 1   3  12   18 cd 2 1  9 cd 2 2  2 2 1   3 cd 2 2  6 cd 2 1   0
In the presence of triaxiality effect only, the corresponding equation is obtained as
 6 1  3 2  2  17   6 1  18 1  9 2  3 2   2  6  16   12 1  18 1
4  9 2  6  6 2   15   2  6 1  6 1  3 2  3 2   14  413
  2  3 2   6 1  12   12 1  6 2   1  3 2   6 1  0
(3.96a)
Case 2. Position of L2 1     2  (see Fig. 3.1 (b))
Let 2    2 ;   1  1  2    1  2  ; 1  1  2 ;
 i  0 i  1,2
2  2 with
(3.97)
Substituting equation (3.97) in equation (3.94), yields
69
210   6  3  29    2  13  2cd 2  9  6cd 2 1  3cd 2 2  28   2 2  24cd 2 1  1  6 cd 2 1  2 cd 2  22  8cd 2
3 cd 2 2 12cd 2 2   3  27   3 3  18 cd 2 1  36cd 2 1  9 cd 2 2  6 cd 2  18cd 2 2  12cd 2  18  12  26
  9  12cd 2 2  2 cd 2 18 cd 2 1  8   2   3  9 cd 2 2  24cd 2 1  6cd 2  25   6 cd 2  3 3  16  6 cd 2 1
18 2  3cd 2 2  6cd 2 1  3 cd 2 2  2 4   6 cd 2 1  24 2  3 cd 2 2  15  6  cd 2  3 3  23   5  2  cd 2
18 cd 2 1  9 cd 2 2   3  12 2  2 2   18 cd 2 1  9 cd 2 2  2 2  2  6 cd 2 1  3 cd 2 2  0
(3.98)
In the presence of triaxiality effect only, the corresponding equation is obtained as
 3 2  61  2  27  18 1  2  9 2  3 2   6 1  6  26   18 1  4  12 1  9 2
6 2   6  25   2  6 1  6 1  3 2  3 2   2 4  423   6 1  3 2   2  2 2
  6 2   12 1  2  6 1  3 2   0
(3.98a)
Case 3. Position of L3   1  (see Fig. 3.1 (c))
Let the distance of the point L3 from the bigger primary be 1  3 .
Since 2  1  1  1    1  3 ; 2    2  3 and
  3    1; 1  1  3 ; 2  2  3 with  i  0 i  1,2
(3.99)
Substituting equation (3.99) in equation (3.94), we obtain
310   3  14  39     2  6cd 2 1  41  3cd 2 2  81  2cd 2  38   2 cd 2  72cd 2 1  240 
6 cd 2 1  247  14  2  24cd 2  3 cd 2 2  36cd 2 2   3  37  11 3  404  22  cd 2  372cd 2 1
66 cd 2 1  186cd 2 2  84  2  124cd 2  33 cd 2 2  784   36   255  306  cd 2 1  277  2  51 3
1080cd 2 1  153 cd 2 2  540cd 2 2  102  cd 2  358cd 2  1562   35   387  cd 2 2  131 3
1926cd 2 1  1950   262  cd 2  963cd 2 2  248  624cd 2  542  2  774  cd 2 1  34
  636  2  1080cd 2 2  2160cd 2 1  616  205 3  1507   410  cd 2  1146  c 2 1  656cd 2  573 cd 2 2  33
  480  673  398 cd 2  990  cd 2 1  432  2  384cd 2  744cd 2 2  1488cd 2 1  199  3  495  cd 2 2  32
 158 2  231 cd 2 2  462  cd 2 1  288cd 2 2  144  224  cd 2  128  112  3  96cd 2  576cd 2 1  3
  56 cd 2  96cd 2 1  12   28 2  28 3  90 cd 2 1  48cd 2 2  45 cd 2 2   0
(3.100)
70
In the presence of triaxiality only, the corresponding equation is obtained as
 6 1  3 2  2  37   66 1  2  33 2  22  6 1  3 2   36   306 1  20  60 1
153 2  102  30 2  )35   82  256  246 1  387 2  774 1  123 2   34 
1152 1  528 1  180  264 2   576 2  368 33   309 2   288  618 1  1008 1 
504 2  230  32   372 1  96  186 2   240 2  168  480 1  33   45 2   56 
48 2  96 1  90 1   0
(3.100a)
It should be noted that some of the above equations have more than one positive root but
in each case there is only one physically acceptable root.This is confirmed by Yamada
and Asada (2011) in the absence of triaxiality. It is also here pointed out that we have not
considered the higher order relativistic corrections because i (i  1, 2),
1
<<1.
cd 2
3.5.3 Stability of collinear points
The stability of an equilibrium configuration is examined, that is its ability to restrain
the body motion in its vicinity. To do so we displace the infinitesimal body a little from
an equilibrium point with small velocity. If its motion is rapid departure from vicinity of
the point, we call such a position of equilibrium an unstable one. If the body oscillates
about the point, it is said to be a stable position.
In order to study the stability of the collinear points, we consider the characteristic
equation:
( p1q2  p2 q1 ) 4  ( p1q6  p5q2  p3q4  p6q1  p2q5  p4q3 ) 2  p5q6  p6q5  0
where,
71
(3.101)
p1  1  W0 , p2  W0 , p3  W0  W0  0, p4  W0  2n  W0 ,
0
0
0
p5  W0 , p6  W0 , q1  W0 , q2  1  W
  , q3  2n  W  W ,
q4  W0  W0  0, q5  W0 , q6  W0 .
The second order partial derivative of W are denoted by subscripts. The superscript 0
indicates that the derivative is to be evaluated at the collinear equilibrium points
 0 , 0  under consideration.
In order to study the stability of the collinear points, it requires to study the motion in
the proximity of these points, hence in this case the second order derivatives evaluated
at  0 , 0  are
W0  1 

3(1   )(   ) 2


5
1
(1   )

3
1

3 (  1   ) 2

5
2


 23
3  15 (2 1   2 ) (  1   ) 2

  3 1   2  
2 
2  27

3 (2 1   2 )
1 
3 2  9(1   )(   ) 2 3(1   ) 3 9 (  1   ) 2  2


(1


)

3


 3
  


2  25
cd 2 
2
2 15
2 13
2 2
2  25


 3(1   )(   ) 3 (  1   ) 
3(1   ) 3 4(1   ) 2 (   ) 2 (1   ) 2 4  2 (  1   ) 2
2 





 
3
3
1
2
1
2
16
14
 26





 7(   ) 7(  1   ) 7  3(   ) 2 1
2
1 3(  1   ) 2 
3(   ) 2
  (1   ) 

 
 3 3
 
4
3
3
5
5
2
1
2
2
1
1  2
2
15  2


2(   )(  1   )
13  23

1
13  2

3(  1   ) 2
1  25

1
1  23

3  3  2  (   ) 2  3  2  3 1  3  (  1   ) 2


2 15
2 13
2  25
1  3  


2  23
(3.102)
W0  1 


(1   )
3(1   )
1

3
1

3
2



 23
 3(1   ) 3  2
3  3 (3 2  4 1 )
1 
1

  3 1   2  
 2   (1   )  3   2   
 3 
5
2 
22
cd 
2
2 13
22 


(1   ) 2
14

7  1
2
1 
 (1   )
1
1

  (1   )    3  3    3 
 3 
4
3
2
1
2
1  2 1  23
 2  1  2 

 3  2   1  3  
2 13
2  23


(3.103)
72
W0  0
W0 
(3.104)
1  1 2 3(1   ) 3 
 
  
cd 2  2
1
2 
0
W
 
1
cd 2
(3.105)
 3 2 3(1   ) 3 
 
  
1
2 
2
(3.106)
W0  0
(3.107)
W0  0
(3.108)
W0  0
(3.109)
Now it will be shown that the discriminant  of equation (3.101) is positive at the
collinear points Li i  1,2,3
To show  is positive it is noticed that
T  4( p1q2  p2 q1 )( p5 q6  p6 q5 )  0
(3.110)
as shown below
T can also be written as


T  4W0 W0 1  W0 1  W0

(3.111)




0
From equations (3.105) and (3.106) it clear that 1  W0  0 and 1  W  0
0
0
Now we will study the signs of W and W at the collinear points Li i  1,2,3
73
Firstly we will do this at L1 , since the coordinate of this point is 1  1   ,0  , then
0
0
1  1  1 and 2  1 where 0  1  1, hence we can write W and W as a function
in 1 say h(1 ) and
f (1 ), respectively. Therefore, in this case from (3.102),
h(1 )  h(0 )   and from (3.103), f (1 )  f (0 )  , hence W0  0 and W0  0
0
0
in which W W  0 and consequently T  0. Hence the discriminant of the equation
(3.101) is positive, and the characteristic roots can be written as 1,2   , 3,4  i
where  and  are real.
Thus 1,2 are real and 3,4 are pure imaginary, hence the motion around the collinear
point L1 is unbounded and the solution is unstable.
Similarly, it can be shown that the points L2 , L3 are also unstable.
CHAPTER FOUR
PERTURBATIONS IN CORIOLIS AND CENTRIFUGAL FORCES,
OBLATENESS AND TRIAXIALITY OF THE PRIMARIES
4.1 Introduction
In this chapter the locations are determined and the stability of the triangular points
taking into consideration the asphericity of the primaries and small perturbations in
Coriolis and centrifugal forces is studied.
74
4.2 Triangular Points with Perturbations in Coriolis and Centrifugal Forces
with the Bigger Primary as an Oblate Spheroid
In this section, the locations are obtained and stability region of the triangular points
when the bigger primary is an oblate spheroids with small perturbations in the Coriolis
and centrifugal forces is also examined.
4.2.1 Equations of motion
The small perturbations in the centrifugal and Coriolis forces are introduced by
parameters   1   1 ;  1  1,  1   2 ;  2  1, and oblateness of the bigger
primary by the parameter A1 
AE 2  AP 2
 1 (McCuskey, 1963) where AE and
5R 2
AP are the equatorial and polar radii of the bigger primary, and R is the distance
between the primaries. Neglecting second and higher powers of A1 , the equations of
motion can be written as:
  2 nd 
W d  W 
 

 dt   
(4.1)
  2 nd  
W d  W 
 

 dt   
 
1  3 
(1   ) 
3
1
W   1  A1  ( 2   2 ) 
 2
1  2 A1  
2  2 
1  2 1   2 cd
 3 1
  3  2
2
  2 1  3  (1   )  1  4 A1  (   )

 


1
3 


 3 
   2   2   2  A1       1  A1  ( 2   2 ) 
8
2
2









2

     2

3  1  
3
3 
3

2
2
2 
 
 
1  2 A1          2  A1       1  2 A1  (   ) 
2  1  2 1   2  
2 

 2 1 





2

A1   2 
A  1 
1  (1   ) 2 
 7 21    1 
 
1  2   2    (1   )   4  3 A1      A1    1  12   

2
2  1  2 1   2 
 2 4    1  2 1   2 



  1     3 A   1
A1  1  3  
1  3A 
 1  
 1  1  2  3  3   1  1  


 
1  2  
2
2 
2  
2   1  2 2 1
1  2 1  2  2  
 1
(4.2)
75
and nd , the perturbed mean motion of the primaries is given by
nd  1 
3
3  1

A1 
1   (1   ) 
2 
4
2cd  3

(4.3)
4.2.2 Locations of the triangular points
The libration points are obtained from equations (4.1) after putting         0.
These points are the solutions of the equations
W
W
0
with     0.


That is, substituting the values of  1  1 and   1   2 in the above equations and
neglecting second and higher orders terms of  1 ,  2 , A1 and their products, the following
system is obtained as
76

(1   )(   )  (  1   ) 3  (1   ) (   ) 
1

 A1  
  1  2
13
 23
2 
15
c

d

  (1   )  1
2
2
   (   )
 3 1 
3
2



 (1   )(   )  (  1   )   1    
3
(1   ) 2 (   )  2 (  1   )
 ( 2   2 ) 


3




 

2
13
 23
2 
14
 24

  1
 7  1 1  7  (   ) (  1   )  3 2   (   ) (1   )(  1   )  (  1   )
  (1   )        


  

5
13
 23
 25
1 23
 2  1  2  2 
 2  1

(   )  3  2  (   ) 1  3  (  1   ) 

  (1   ) 
2
2
 3


  1 3 1 
   (   )
3
3
1  2
2 1
22
3




 (1   )(   )  (  1   ) 
 1     
3
 7  1 1 
 ( 2   2 ) 

     (1   )1    
  3 
3
3

2
1
2
 2  
 2  1  2 

 1

7  (   ) (  1   )  3 2   (   ) (1   )(  1   )   (   ) (  1   )  3  2  (   )
  




  
 3
5
2 
13
 23
 25
1 23
2 13
 2  1
  1  2

1  3  (  1   )    A  9 1   (1   )    3 ( 2   2 )    9(1   )(   )  9(1   )(   )

 3(1   ) 9(1   ) 9 
9  (  1   )  2
2


 (   )  
3
3
42
2 1
22

 2 1
2  23

 
1
 
 4
3


2


4 15
4 13

 7 21  1 1 
2(1   ) 2 (   ) 


   (1   ) A1  3    

6
1

 4 1 4  1  2 

7  3(   )  21  (   ) (  1   )  9 2   (   ) (1   )(  1   )  3(1   )(   )
  


  
  

5
2 
2 15  4 
13
 23
 25
2 15
 4  1


3(   ) 3(  1   ) 3 (   ) 3(1   )(   ) 3(1  3 )(  1   ) 




  0
2 13  2
2 1  23
4 13
2 13
4  23

and
F  0,
(4.4)
where,
77
 1   3  1  
 1
F  1  3  3  A1 1  5   1   2
1
2 2 
1 

 cd
1   
   (1   )  1 2
2
 
  (   )  3 
 3 1 
3  2
2 
 
 1
 7  1
 1      (1   ) 2  2 
3
1    1    3   (1   ) 
 ( 2   2 )  3  3   
 4    (1   )     3  3    3  3    2  5 

4
2
 2   1
2 
 2  2  1
25 
 2  1  2   1
 1

 3  2   1  3     3 1  1  (1   )   ( 2   2 )   1       3 ( 2   2 )  1  
1
1
 3 
 1 


 3

3
1 2 1  2
2 13
2  23 
2  2

  3
 1
 1

 7  1
 3  2 
  
1    1    3   (1   )   1
1
  (1   )1    3  3    3  3    2  5 

 3 
3 
5
3
 2  
 2  2  1
 2   1 2 1  2
2 13
 2  1  2   1

1  3     A  9 1   (1   )   3 ( 2   2 )   9(1   )  9(1   )  9  ( 2   2 )  3(1   )  9(1   )
 1  


3
5
3
3 
3

9 2(1   ) 2 
1
1  3   1   9 2   1   3  1 ) 1 
 21  1

   (1   ) A1     5  3  3    3  3     5  5    5  3
6
22
1 
 2  4  1
 2  2  1
1
 4  1 1  2  2  1

1
1
 (1  3 )  

 3
 
3
  2 1  2 2 1
2  23  
22
 4
 
3
 2

4 1
4 1
42 
2 1
2 1
3
1
The triangular points are the solutions of equation (4.4) with  0. Since
in the case
i.e. 1   2
1
 1 and
cd 2
1
 0 and in the absence of small perturbations and oblateness
cd 2
 A1  0 , one can obtain 1  2  1; it is assumed in the relativistic R3BP
that 1  1  x and 2  1  y where x, y  1, may be depending upon relativistic,
perturbations and oblateness factors. Substituting these values in the equations (3.6),
solving them for  ,  and ignoring terms of second and higher powers of x and y , the
solutions are
  x y
1  2
2
(4.5)
 3 x y


2
3 

   
78
Substituting the values of 1, 2 ,  ,  and neglecting the second and higher orders terms
in x 2 , y 2 ,
x y
,
, 1 ,  2 , A1
cd 2 cd 2
in equations (4.4) with  0 , the following system is
obtained as
3
3  5 3    3 3 
3
1
A1  y 
A1  2
 1     A1  
 x   
2  2 2   2
2 
4
cd
2
 9  27  2 9 3 


 

16
8 
 16
 11 125 183 2 63 3 
 8 3  18 2  17   4  



A


 1 
 1   0
32
32
8 
16
8

 
3 A1 1
15


 2
3 1     1    A1  x  3 y 
2
2
cd


 21    2   11 53 49 2 3 3 
 



 A1
8
16
2 
 4 16

 5 2  5  2  

 1   0
2

 
(4.6)
Solving these equations for x and y , the solutions are
x
  2  3   44  51  22 2  30 3 
 8 3  8 2  7   8 

A


 1 
 1 ,
8cd 2
48(   1)cd 2
24(   1)cd 2




1    5  3 
y
8cd 2
 1 74  86  51 2 
 8 2  28  27 
 
A



 1
1
48cd 2
24cd 2
2



(4.7)
Thus, the coordinates of the triangular points  ,  denoted by L4 and L5 respectively
are,
2
3
 18 2  33  14 
1  2 
5   1  30  109  115  21  



1



A

,


 1  24(   1)cd 2  1
2  4cd 2   2
48(   1)cd 2




3  118  211  159 2  81 3  
1
3
 3
2 

    1 
5  6   6     

A1 
2 
2


2
12
c
6
144(


1)
c
d
d

 



3  10 3  32 2  51  30   
 2 3 6  2  6   5

 1 
 


2
2
 
54 cd
72(   1)cd
 9
 
(4.8)
79
4.2.3 Stability of L4
In this section, the same methodology as in section (3.2.3) is used.
Let (a,b) be the coordinates of the triangular points L4
We set   a   ,  b   , ( ,   1) in the equations (4.1).
First, the terms on their R.H.S, are computed, neglecting second and higher order terms,
the followings are obtained as
 W 


 A  B  C  D



  a  , b 
(4.9)
where,

226  1036  1758 2  1056 3  87 4  

3
1
2   3  8  9 
A  1 
 2  19  19    8 
 A1
2
4  2cd 2
32(


1)
c
d
 

 5 31  168  264 2  126 3 
 
 1 ,
8(   1)cd 2
4

3  46  393  599 2  135 3  417  4  

3 3
2   3  26  19 
B

1  2  1  2   
 A1
2
4
3
c
8
96(


1)
c
d  
d



11 3(1  2 )
3(34  279  635 2  528 3  132 4 ) 


 1 ,
12
144(   1)cd 2


C
 3  46  35 
3
 5 3(1  2 ) 
 3(1  2 ) 
1  2    
 A1  
 1  
2 ,
2 
2
2
2
2cd
24
c
18
c
2
c




d
d
d







 4  15  15 2 
 6  5  5 2 
6  5  5 2  22  33  45 2 
1  
 2 .
D

 A1  
2c 2
8c 2
6c 2
2c 2






Similarly,
80
 W 


 E  B1   C1  D1 
    a  , b 
(4.10)
where,
3  46  393  599 2  135 3  417  4  

3 3
2   3  26  19 
E

1  2  1  2   
 A1
4
8
96(   1)cd 2
 3 cd  

3(34  279  635 2  528 3  132 4 ) 
11 3(1  2 )


 1 ,
12
144(   1)cd 2



290  1292  2028 2  1221 3  111 4  

9
7
2   33
B1  1 
 2  3  3    8 
 A1
4  6cd 2
96(   1)cd 2
 

 7 55  168  216 2  102 3 
 
 1 ,
8cd 2
4

  20  13  9 2  
 2     2 
 4     2 
1
2
C1 
 4       
 A1  
 1  
2,
2cd 2
8cd 2
2cd 2
2cd 2






D1  
 5 3(1  2 ) 
  3(1  2 ) 
3 1  2   3  46  35 

 A1  
 1  
2.
2
2
2
2
2cd
24
c
18
c
2
c




d
d
d






d  W 
 F  B2   C2  D2 
 
dt    a  , b  
(4.11)
where,
F
 3  46  35 
3
 5 3(1  2 ) 
 3(1  2 ) 
1

2





 A1  
 1  
 2 ,
2
2
2
2
2cd
24cd


 18cd

 2cd

  20  13  9 2  
 2     2 
 4     2 
1
2
B2  
4






A






 1 
 1 
2,
2
2
2
2cd 2
8
c
2
c
2
c
d
d
d






81
  23  10  6 2  
 1    2 
 10     2 
1
2
C2 
17  2  2   
 A1  
 1  
2,
2
4cd 2
8cd 2
2cd 2
 2cd





D2  
 3 14  13 
3
 3(1  2 ) 
 3(1  2 ) 
1

2





 A1  
 1  
2.
2
2
2
2
4cd
24cd


 9cd

 2cd

d  W 


 A3  B3   C3  D3 
dt     a  , b 
(4.12)
where,
  22  33  45 2  
 4  15  15 2 
 6  5  5 2 
1
2
A3 
6

5


5


A




 
 1 
 1 
2,
2
2
2
2cd 2
8
c
6
c
2
c
d
d
d






B3  
 3  46  35 
3
 5 3(1  2 ) 
 3(1  2 ) 
1  2   
 A1  
 1  
2 ,
2 
2
2
2
2cd
24cd
18cd


 2cd



C3  
 3 14  13 
3
 3(1  2 ) 
 3(1  2 ) 
1  2   
 A1  
 1  
2 ,
2 
2
2
2
4cd
24cd
 9cd

 4cd



3  5  2  2 2    25  30  18 2  
 7  3  3 2 
 3(1  2 ) 
D3 

A


 1 
2.
 1  
2
2
2
2
4cd
8
c
6
c
4
c


d
d
d





Thus, the variational equations of motion corresponding to equation (4.1), on making
use of equation (4.3), can be expressed as
p1  p2   p  p4   p5  p6   0,
(4.13)
q1  q2   q3  q4   q5  q6   0.
where,
82
 3

3  1



p1  1  C2 , p2  D2 , p3  F  C , p4   B2  2 1  A1   2 
1   (1   )  1   2    D  ,
2 
2cd  3

 4



p5   A , p6   B,
 3

3  1

q1  C3 , q2  1  D3 , q3  2 1  A1   2 
1   (1   )  1   2    C1  A3 , q4  B3  D1 ,
2 
2cd  3

 4

q5   E , q6   B1
Then, the corresponding characteristic equation is
( p1q2  p2 q1 ) 4  ( p1q6  p5q2  p3q4  p6q1  p2q5  p4q3 ) 2  p5q6  p6q5  0 (4.14)
Substituting the values of pi , qi , i  1, 2,...,6 in equation (4.14) and neglecting second
and higher powers of small quantities, the characteristic equation (4.14) becomes
4  b2  d  0
(4.15)
where,
80  108  105 2  18 3  



9   3
68  25  25 2 
b  1  2     3 
 A1  3 
 1
8cd 2
4cd 2
 cd   2



 147  30  30 2 
 8 
2 ,
2cd 2


2
3
27  (1   ) 9  65  77   24  12  117  1   
d



4
8cd 2
4

3  80  7245  9624 2  3366 3  846 4  
 A1 
64cd 2

 33 (1   ) 3 (1867  2082  540 2  336 3 
 243  324 2  162 3  81 4 




 1 
2.
2
32cd 2
4cd 2




83
For
1
 0 and in the absence of small perturbations in the centrifugal and Coriolis
cd 2
forces and oblateness (i.e.  1   2  A1  0), equation (4.15) reduces to its well-known
classical restricted problem form (see e.g. Szebehely, 1967a):
4  2 
27
 (1   )  0.
4
The discriminant of equation (4.15) is
 54 1269
 108 5013
126
81 
405
126 
 2 
A1  2 1  2  2   4   2 
A1 
  2  2   3   27  117 A1  661
2
2
2 1
8cd
cd
cd
8cd
2cd
cd
 cd

 cd



7113
3073
354
693 
20439
5501
273
585 
A1 
  2  2  2   2   27  111A1  661 
A1 
  2 2  2  
2
2 1
2
2 1
4cd
4cd
cd
2cd 
16cd
8cd
cd
2cd 

1  3 A1  61  16 2 
8
88
291
18
A  2 1  2  2  2 .
2 1
cd
cd
cd
cd
(4.16)
Its roots are
2 
b 
2
(4.17)
where,
80  108  105 2  18 3  



9   3
68  25  25 2 
b  1  2     3 
 A1  3 
 1
8cd 2
4cd 2
 cd   2



 147  30  30 2 
 8 
2 ,
2cd 2


From (4.16), we have
 108 5013
d   54 1269
126
81 
405
162 
 4  2  2 A1  2 1  2  2   3  3  2  2 A1  2 1  2  2   2  2  27  117 A1  661
d
8cd
cd
cd 
cd
2cd
cd
 cd
 cd


7113
3073
354
693  
20439
5501
273
585 
A  2 1  2  2  2     27  111A1  661 
A1  2 1  2  2  2   0
2 1
2
4cd
4cd
cd
2 cd  
16cd
8cd
cd
2 cd 
84
 1
for    0, 
 2
(4.18)
But
    0  1 
    1  
2
18
8
88
291
 3 A1  2 A1  61  2 1  16 2  2  2  0
2
cd
cd
cd
cd
23 207 117
32585
4115
3645


A1 
A
  16 2 
2  0
2
2 1
2 1
4 4 cd
4
128cd
16 cd
16cd 2
(4.19)
 54 1269
 108 5013
d 2
126 
405
162 
 12   2 
A1  2 1   2  6  2  2 A1 
  2 2  2 
2
2
2 1
d
8cd
cd
cd
2cd
cd
 cd

 cd


7113
3073
354
693 
 1
2  27  117 A1  661 
A





 0     0, 
1
1
2
2
2
2
2 
4cd
4cd
cd
2cd 
 2

(4.20)
This implies that
d
is monotonic increasing in
d
 1
 0, 
 2
But
 d 
5501
273
585 20439
 1
  2 2  2 
A1  0     0,  (4.21)

  27  661  111A1 
2 1
2
8cd
cd
2cd
16cd
 2
 d    0
 d 
33
52143
 
A1  6 A1

 1
2 1
4cd
16cd 2
 d   
(4.22)
2
In order to study the monotonicity of  , two cases are considered
 d 
Case 1: 
 0
 d    1
2
Hence for this case, the table of variation of  is given below
85
Table 4.1 : Variation of 

0
1/2
d 2
d2

d
d


   0
   1 2
 1
From the above table it can be seen that  is monotone decreasing in  0,  . Since
 2
    0
and  
  1
are of opposite signs, and is monotone and continuous and by the
2
 1
intermediate value property there is one value of  say. c in  0,  for which   0 .
 2
 d 
Case 2: 
0

 d    1
2
Since from equation (4.20),
d
is monotone increasing in
d
 d 
 1
 0,  and 
 0
 2
 d    0
 d 
 1
0
and 
 1  0 , this implies that there xists    0, 2  such that


 d   
2
hence
d
1
d

 0     0 ,  .
 0     0,  0  and
d
2
d

Hence, the following table of variation is given below.
86
 d 
 0,


 d    0
Table 4.2: Variation of 

0
0
12

d
d
 d  d   1/ 2

 d  d    0

    0
  1
2
    
0
Since    0  0 and     1  0 , it can be concluded from the above table that
2
  
0
 0 , hence     0 and
    
0
are of opposite signs and  is monotonic
 1
decreasing and continuous in  0,  and by the intermediate value property there is one
 2
value of  say.   in  0,  0  for which   0 . Hence c  c  c .
Solving the equation   0 , using equation (4.16), the critical value of the mass
parameter is obtained as
 19733  15493 69 
4  36 2  191 
1 1
17 69 1 
13 
69 
 1 
A1  

 A1 

2
2
2 18
486cd
9
536544cd
69 
27 69


c  
 34155  175301 69 
 47 69 

 1  
2
2 
2
804816 cd


 81cd 
(4.23)
There are three possible cases regarding the sign of the discriminant 
87
When 0    c ,   0, the values of 2 given by equation (4.17) are
i.
negative and therefore all the four characteristic roots are distinct pure
imaginary numbers. Hence, the triangular points are stable.
When  c   
ii.
1
,   0 , the real parts of the characteristic roots are
2
positive. Therefore, the triangular points are unstable.
iii.
When    c ,   0 , the values of 2 given by equation (4.17) are the same.
Hence the solutions contain secular terms. This induces instability of the
triangular points.
Hence, the stability region is
0    0 
 19733  15493 69 
4  36 2  191 
17 69 1 
13 
 1 
A1  

 A1 

2
2
486cd
9
536544cd
69 
27 69


 34155  175301 69 
 47 69 

 1  
2
2 
2,
804816 cd


 81cd 
(4.24)
where  0  0.03852... is Routh‟s value
Equation (4.24) can be written as:
0    0  p
(4.25)
with
p
 19733  15493 69 
4  36 2  191 
17 69 1 
13 
 1 
A1  

 A1 

2
2
486cd 9 
536544
c
69 
27
69
d


 34155  175301 69 
 47 69 





1
2 

 2.
804816 cd 2


 81cd 
88
4.3 Triangular Points with Perturbations in the Coriolis and Centrifugal
Forces with a Triaxial Bigger Primary
In this section, the locations are obtained and stability region when the bigger primary is
triaxial with small perturbations in the Coriolis and centrifugal forces is also examined.
4.3.1 Equations of motion
The small perturbations in the centrifugal and Coriolis forces and triaxiality of the
bigger
primary
are
introduced
with
the
help
of
  1  1; 1  1,   1   2 ;  2  1 ,  i (i  1,2) with  i  1, respectively,
parameters
where
h2  f 2
b2  f 2
1 
, 2 
. (McCuskey, 1963) with h, b, f as lengths of its semi-axes
5R 2
5R 2
and R is the dimensional distance between the primaries. As Katour et al. (2014), the
triaxiality coefficients are not included in the relativistic part since the magnitude of
those terms is so small due to cd 2 , where cd is the speed of light. Consequently,
ignoring second and higher powers of  i , the equations of motion can be written as:
  2 nd 
W d  W 
 

 dt   
(4.26)
  2 nd  
W d  W 
 

 dt   
89
3 1    2
1 3
(1   )  1  

W  1   2 1   2   ( 2   2 ) 


2







 2   1 
1
2
2 2
1
 2 2 13
2 15


1
cd 2
 3 1
1
 2
2
2
2
2
2

  2 1  3  (1   )  (   )  8      2     (   )








2
3  1   
1  (1   ) 2  2 
 
    2   2  2     ( 2   2 )  
 2
2  1
2 
2  12
2 







 1 3  2 1  3  
7
1   2   1  

 1
  (1   )  4          3  3    


  ,
2   1  2  2  1
2 
2 1
22  


 1  2

(4.27)
and nd the perturbed mean motion of the primaries is given by
nd  1 
3
3
1
 21   2   2 1   (1   ) 
4
2cd  3

(4.28)
4.3.2 Locations of the triangular points
The libration points are obtained from equation (4.26) after putting         0.
These points are the solutions of the equations
W
W
with     0.
0


Substituting for simplicity,   1   1 and   1   2 in the aforesaid equations and
neglecting second and higher powers of  i ,  i i  1,2 , and also their products, the
following system are obtained
90

3(1   ) (   )(2 1   2 )
(1   )(   )  (  1   ) 
3 

  3 1   2   
3
3
1
2
2 
2 15


3 2
  (1   )  1
2
2
2  (1   )(   )
   (   )  (   ) 
 3 1 
3  2
2
13




15(1   ) (   )( 2   1 ) 2
1
 1  2
7
2 1
cd

 7  1 1 
 (  1   )   1     (1   ) 2 (   )  2 (  1   )
  

  (1   )    
  3
3
4
4
2
2 
1
2
 2  1  2 
  1
7  (   ) (  1   )  3 2   (   ) (1   )(  1   )  (  1   ) (   )
  


 3
  

5
2 
13
 23
 25
1 23
1  2
 2  1

 3  2  (   )  1  3  (  1   )    3 1   (1   )    ( 2   2 )  3 ( 2   2 )  (1   )(   )

 1



2 13
2  23
3 
2
13





 7  1 1  7  (   ) (  1   ) 
 1     
 (  1   ) 
     (1   )1        

  3 

3
2
 2  
13
 23

 1

 2  1  2  2 
3 2   (   ) (1   )(  1   )   (   ) (  1   )  3  2  (   ) 1  3  (  1   )  
 




   0
 3
2  15
 25
1  23
2 13
2  23
  1  2
 
and
F  0,
(4.29)
where,
2
 1    
3  3(1   )  3
 15(1   )  2   1 
F  1  3  3    3 1   2  


2


 1
1
1
2  
2 
15  2 2
2 17



1
cd 2
1    3 2
   (1   )  1 2
   (1   ) 2  2 
2
2 1 

3
1


(



)

3


(



)

 4


 3


 
3
2  2
 23   14
2 
 2
 
 1
 1
 7  1
 3  2   1  3   
1    1    3   (1   ) 
1
1
  (1   )     3  3    3  3    2  5 
 3 


5
3
 2  2  1
 2  1 2 1  2
2 13
2  23 
 2  1  2   1



1 3 1 

 7  1
 1    3 2
 1     
1 
2
2
   (   2 )  3  3     (1   )1     3  3 
  (   )  
2  2
 2  

 2  1  2 
 1
 1
 (1   ) 
3
  1    3   (1   )   1
 3  2   1  3   
1
 3  3   2  5 
 3 


5
3
 2  2  1
 2   1  2 1  2
2 13
2  23  
 1
91
The triangular points are the solutions of equations (4.29) with  0. Since
and in the case
1
 1
cd 2
1
 0 and in the absence of small perturbations and triaxiality
cd 2
i.e. 1   2   2   1  0 ,
one can obtain 1  2  1; we assume in the relativistic
R3BP that 1  1  x and 2  1  y where x, y  1, may be depending upon
relativistic, triaxiality, centrifugal and Coriolis factors. Substituting these values in the
equations (3.6), solving them for  ,  and ignoring terms of second and higher powers
of x and y , and their products, the solutions are
  x y
1  2
2
(4.30)
 3 x y
   


3 
 2
Substituting the values of 1, 2 ,  ,  from the above in equations (4.29) and neglecting
second and higher powers of x , y , 1, 2 , 1,  2 , the following system is written as
 57 2  69 1    1
3
3
45
1    x  y   1   2  
2
2
16
16
cd 2
 8 3  18 2  17   4  

 1   0
8

 
21
3
1
3 1    x  3 y    1   2    1  3 2    2
8
8
cd
 9 27  2 9 3 


 

16
8 
 16
 21    2   5 2  5  2  


 1   0
8
2

 

(4.31)
Solving these equations for x and y , the solutions are
x
  2  3  11
 8 3  8 2  7   8 








 1 ,
2
1
8cd 2
8
24(   1)cd 2


1    5  3 
y
8cd 2
 8 2  28  27 
 1 3
 1


 1   
 1  2  
 1.
24cd 2
 2 2 
 2 


92
(4.32)
Thus, the coordinates of the triangular points  ,  denoted by L4 and L5 respectively
are,
 18 2  33  14 
 1 3
1  2 
5  1 1 

  2  
1 
 
 1 ,
1  
2
2  4cd 2   8 2 
 2 8 
 24(   1)cd

 3 
 19 1    2 3
1
2  1 23 
2

5

6


6








1 


 1  
  2   
2
3  2 8 
 8 2    9
 2  12 cd
  
3  10 3  32 2  51  30   
6 2  6  5
 1 


 
54 cd 2
72(   1)cd 2
 
(4.33)
4.3.3 Stability of L4
Let  a, b  be the coordinates of the triangular point L4 .
Setting   a   ,  b   , ( ,   1) in the equations (4.26).
The terms of their R.H.S. are computed, neglecting second and higher order terms, the
following are obtained as
 W 


 A  B  C  D



  a  , b 
(4.34)
where,
3 15 2  19  8 
3  31 2    8 
3
1
2 
A  1 
1 
2
 2  19  19  
4  2cd 2
16
16

 5 31  168  264 2  126 3 
 
 1 ,
2
4
8(


1)
c
d


93
3  89 2  47   8 
3  37  2  9  8 

3 3
2 
B
1 
2
1  2  1  2  
4
3
c
16

16

d 

11 3(1  2 )
3(34  279   635 2  528 3  134 4 ) 


 1 ,
12
144(   1)cd 2


C
 5 3(1  2 ) 
 3(1  2  ) 
3
1

2









2 ,
1
2
2
2cd 2
 18cd

 2cd

D
 6  5  5 2 
6  5  5 2  4  15  15 2 




 1 
 2.
2
2
2cd 2
6
c
2
c
d
d




Similarly, we obtain
 W 
 E  B1  C1  D1


   a  , b 
(4.35)
where,
3  89 2  47   8
3  37  2  9  8 

3 3
2 
E
1 
2
1  2  1  2  
4
16
16
 3 cd 
11 3(1  2 )
3(34  279  635 2  528 3  132 4 ) 


 1 ,
2
12
144(


1)
c
d


3 15 2  29  8 
3 15 2  7   8 
9
7
2 
B1  1 
1 
2
 2  3  3  
4  6cd 2
16

16


 7 55  168  216 2  102 3 
 
 1 ,
2
4
8
c
d


C1 
 2     2 
 4     2 
1
2

4








  2c 2  1  2c 2   2 ,
2cd 2
d
d




D1  
3 1  2   5 3(1  2 ) 
  3(1  2 ) 

 1  
2.
2
2
2cd
18cd
2cd 2




94
d  W 
 F  B2   C2  D2 


dt    a  , b  
(4.36)
where,
F
 5 3(1  2 ) 
 3(1  2  ) 
3
1

2









2 ,
1
2
2
2cd 2
 18cd

 2cd

 2     2 
 4     2 
1
2
B2  
 4        2c 2  1   2c 2   2 ,
2cd 2
d
d




C2 
 1    2 
 10     2 
1
2
17

2


2





  2c 2  1  2c 2   2 ,
4cd 2
d
d




D2  
 3(1  2 ) 
3
 3(1  2  ) 
1  2   
 1  
 2.
2 
2
2
4cd
 9cd

 2cd

d  W 


 A3  B3   C3  D3 
dt     a  , b 
where,
A3 
 4  15  15 2 
 6  5  5 2 
1
2
6

5


5







 1 
2,
2cd 2
6cd 2
2cd 2




B3  
 5 3(1  2 ) 
 3(1  2  ) 
3
1

2










2 ,
1
2
2cd 2
18cd 2 

 2cd

C3  
 3(1  2 ) 
3
 3(1  2  ) 
1  2   
 1  
2 ,
2 
2
2
4cd
 9cd

 4cd

3  5  2  2 2   7  3  3 2 
 3(1  2 ) 
D3 

 2.
 1  
2
2
2
4cd
6
c
4
c


d
d



95
(4.37)
The variational equations of motion corresponding to equation (4.26), on making use of
equation (4.28), can be expressed as
p1  p2   p3  p4   p5  p6   0,
(4.38)
q1  q2   q3  q4   q5  q6   0.
where,


 3

3  1

p1  1  C2 , p2  D2 , p3  F  C , p4   B2  2 1   2 1   2    2  2 1   (1   )  1   2    D  ,
2cd  3

 4



p5   A, p6   B,
 3

3  1

q1  C3 , q2  1  D3 , q3  2 1   2 1   2    2  2 1   (1   )  1   2    C1  A3 , q4  B3  D1 ,
2cd  3

 4

q5   E , q6   B1.
Then, the characteristic equation is
( p1q2  pq1 ) 4  ( p1q6  p5q2  p3q4  p6 q1  p2q5  p4q3 ) 2  p5q6  p6 q5  0
(4.39)
Substituting the values of pi , qi , i  1, 2,...,6 in equation (4.39), the characteristic
equation (4.39) after normalizing becomes
4  b2  d  0
(4.40)
where,


 147  30  30 2 
9 
3
68  25  25 2 
b  1  2   31   2  3  2  3 


 1 8 
 2 ,
2
4cd 2
2cd 2
 cd 




96
2
3
2
9 10  47   37  2 
27  (1   ) 9  65  77   24  12  9  10  99  89 
d


1 
2
4
8cd 2
16
16
 33 (1   ) 3 (1867  2082  540 2  336 3 
 243  234 2  162 3  81 4 




 1 
2.
2
2
2
32
c
4
c
d
d




For
1
 0 and in the absence of small perturbations (in the centrifugal and Coriolis
cd 2
forces) and triaxiality (i.e.  1   2   1   2  0), equation (4.40) reduces to its wellknown classical restricted problem form (see e.g. Szebehely, 1967a):
4  2 
27
 (1   )  0.
4
The discriminant of equation (4.40) is
 54 126
 108 405

81 
162 
801
333
3073
   2  2 1  2  2   4   2 
  2  2   3   27 
1 
 2  661 
1
2 1
cd
cd
2cd
cd
4
4
4cd 2
 cd

 cd



354
693  2 
891
447
5501
273
585 
57
63
2 
   27 
1 
 2  661 
  2 2 
  1 1   2
2
2 
2 1
2 
cd
2 cd 
4
4
8cd
cd
2 cd 
2
2

61 
88
18 291
  16 2  2  2  2
2 1
cd
cd
cd
(4.41)
Its roots are
2 
b 
2
(4.42)
where,


 147  30  30 2 
9 
3
68  25  25 2 
b  1  2   31   2  3  2  3 
 1  8 
 2 ,
2
4cd 2
2cd 2
 cd 




From equation (4.41), the followings are obtained
97
 54 126
 108 405
d
81 
162 
801
333

 4  2  2 1  2  2   3  3  2 
  2  2   2  2  27 
1 
 2  661
2 1
d
cd
cd
2cd
cd
4
4

 cd

 cd



3073
354
693 
891
447
5501
273
585 
  2 2 
   27 
1 
 2  661 
  2 2 
0
2 1
2 
2 1
4cd
cd
2 cd 
4
4
8cd
cd
2 cd 2 

(4.43)
 1
for    0, 
 2
But
    0  1 
    1  
2
57
63
88
291
18
 1   2  61  2 1  16 2  2  2  2  0
2
2
cd
cd
cd
23 207 525
57
4115
3645


1   2 
  16 2 
2  0
2
2 1
4 4 cd
16
16
16 cd
16cd 2
(4.44)
 54 126
d 2
81   108 405
162 
801
333

 12   2  2 1  2  2   6  2 
  2  2    2  27 
1 
2
2
2 1
d
cd
cd
2cd
cd
4
4

 cd
  cd

661 
3073
354
693
  2 2  2
2 1
4cd
cd
2cd
  0     0,

1
2 
(4.45)
This implies that
d
is monotone increasing in
d
 1
 0, 
 2
But
 d 
891
447
5501
273
585
 1
1 
2 
  2  2  2  661  0     0, 

  27 
2 1
4
4
8cd
cd
2cd
 2
 d    0
(4.46)
 d 
33
45
57
  1   2

 1
2 1
4cd
2
2
 d   
(4.47)
2
In order to study the monotonicity of  , two cases are considered:
98
 d 
Case 1: 
 0
 d    1
2
For this case, the table of variation of  is given below.
Table 4.3. Variation of 

0
1/2
d 2
d2

d
d


    0
   1 2
 1
From the above table it can be seen that  is monotonic decreasing in  0,  .
 2
1
are of opposite signs and  is monotone continuous and
2
 1
by the intermediate value poperty there is one value of  say. c in  0,  for which
 2
Since    0 and     
  0.
 d 
Case 2: 
0

 d    1
2
Since from equation (4.45),
 d 
d
 1
is monotonic increasing in  0,  and 
 0
d
 2
 d    0
 d 
 1
0
and 
 1  0 , this implies that there exists    0, 2  such that


 d   
2
99
 d 
 0,


 d    0
d
1
d

 0      0 ,  , hence we have the following
 0     0,  0  and
d
2
d

table of variation of  below.
hence
Table 4.4: variation of 

0
0
12

d
d
 d  d   1/ 2

 d  d    0

    0
   1
2
    
Since
    
    0  0
0
and
    1  0 ,
0
it can be concluded from the above table that
2
 0 , hence since     0 and      0 are of opposite signs, and  is monotonic
decreasing and continuous in  0,  0  and by the intermediate value property there is
one value of  say. c in  0,  0  for which   0 . Hence c  c  c
Solving the equation   0 , using equation (4.41), the critical value of the mass
parameter is obtained as
4  36 2  191 
1 1
17 69 1  5
59 
1  19
85 
69 
  
1   
2 


2
2 18
486cd
2  6 9 69 
2  18 9 69 
27 69
c  
 34155  175301 69 
 47 69 
 
 1  
2
2 
2
804816
c
81
c
d
d




100
(4.48)
c  0 
4  36 2  191 
17 69 1  5
59 
1  19
85 
  
1   
2 


2
486cd
2  6 9 69 
2  18 9 69 
27 69
 34155  175301 69 
 47 69 

 1  
2
2 
2
804816
c
81
c
d
d




(4.49
)
where  0  0.03852... . is the Routh‟s value.
The following three regions of the values of  are consided separately.
When 0    c   0, the values of 2 given by equation (4.42) are
i.
negative and therefore all the four characteristic roots are distinct pure
imaginary numbers. Hence, the triangular points are stable.
ii.
1
When  c    ,   0 , the real parts of the characteristic roots are positive.
2
Therefore, the triangular points are unstable.
iii.
When    c ,   0 , the values of 2 given by equation (4.42) are the same.
Hence the solution contains secular terms.This induces instability of the
triangular points.
Hence, the stability region is
0    0 
4  36 2  191 
17 69 1  5
59 
1  19
85 
  
1   
2 


2
486cd
2  6 9 69 
2  18 9 69 
27 69
 34155  175301 69 
 47 69 
 
 1  
2
2 
2
804816 cd


 81cd 
(4.50)
Equation (4.50) can be written as
0    0  p
(4.51)
101
with
p
4  36 2  191 
17 69 1  5
59 
1  19
85 
  
1   
2 


2
486cd
2  6 9 69 
2  18 9 69 
27 69
 34155  175301 69 
 47 69 
 




1
2 

2
804816 cd 2


 81cd 
4.4 Collinear Equilibrium Points in the Relativistic R3BP with a Smaller
Oblate Primary
In this section, the analytical and numerical locations of the collinear points are obtained
and also their stability when the smaller primary is oblate is examined.
4.4.1 Equations of motion
The effect of oblateness of the smaller primary is included with the help of the
parameter A2  1 and the equations of motion of an infinitesimal mass can be written
as Brumberg (1972) and Bhatnagar and Hallan (1998) as:
  2nd 
W d  W 
 

 dt   
(4.52)
  2nd  
W d  W 
 

 dt   
with
102
W
A  1
1 2 2
1   
nd (   2 ) 
  1  22   2
2
1
 2  2  2  cd
 1 2  2
2
2
2

 8     2nd (   )  nd (   )


2
A 
3  1   
 
 1  22    2   2  2nd (  )
2  1
2  22  

A 
1  (1   ) 2  2 
 
 2  1  22 
2

2
1
2  22 


2


 1 1 
A 
7
   (1   ) nd    nd     1  22  

2
  1  2  2  2  
 

  1  
 1 
A 
A2  3  2 1   
A2  
1
 nd 2  3  3 1  22   2  nd 2 

1  2  
1  2  
2
2  22  
2 1
22  22  2
 1
 1  2  2  2 

A2   
1


2  
 ,
 2  2   
(4.53)
nd 2  1 
3
3  1

A2 
1   (1   ) 
2 
2
2cd  3

(4.54)
12  (   ) 2   2
(4.55)
  (    1)  
2
2
where 0   
2
2
1
is the ratio of the mass of the smaller primary to the total mass of the
2
primaries, 1 and  2 are distances of the infinitesimal mass from the bigger and smaller
primary, respectively; nd is the perturbed mean motion of the primaries; c is the
velocity of light.
A2 
AE

 AP 2
 1. (McCuskey, 1963), where AE and AP are
5R 2
2
the equatorial and polar radii of the smaller primary, and R is the distance between the
primaries.
It should be noted here that the second and higher powers of A2 and 12 have been
cd
ignored in writing above equations
103
4.4.2 Locations of collinear points
Equilibrium points are those points at which no resultant force acts on the third
infinitesimal body. Therefore, if it is placed at any of these points with zero velocity, it
will stay there. In fact, all derivatives of the coordinates with respect to the time are zero
at these points. Therefore, the equilibrium points are solutions of equations
W  0 and W  0
(4.56)
where, W and W may be written as
W   
(1   )(   )  (  1   ) 3   (  1   )  1

 A2  
 2
13
 23
2 
 25
 cd

 1
 1
2
2
 3 1  3  (1   )   2  (   )



 (1   )(   )  (  1   )   1    
3
(1   ) 2 (   )  2 (  1   )
 ( 2   2 ) 


3







2
13
 23
2 
14
 24

  1
 7  1 1  7  (   ) (  1   )  3 2   (   ) (1   )(  1   ) 
  (1   )        


  

5
13
 23
 25
 2  1

 2  1  2  2 

 9   1    
(   ) (  1   )  3  2  (   ) 1  3  (  1   ) 
3 2
2



  A2  1 
   (   ) 
3
3
3
3
1  2
1  2
2 1
22
4
3
2


 
 9  (  1   ) 9(1   )(   ) 9  (  1   )  2
 3 9  9(1   ) 
2  2 (  1   ) 
2


(



)










3
4  25
4 13
4  23
2 1 
 26



 22 22
21  1 1  7  3(  1   )  21  (   ) (  1   ) 
 7
  1    A2  3       

  

2  25
13
 23
 4 

 4  2 4  1  2  2 
9   (   ) (1   )(  1   )  3 (  1   ) 3(   ) 3(  1   ) 3(3  2)(   )
 2 





4  15
 25
2  25
2 13  2
2 1  23
4 13


3(1   )(  1   ) 3 (  1   ) 


4  23
2  23

and
W  F
with
104
F  1
(1   )

3
1

 3 
  1
 A2 1  5   2
3
 2 2   2  cd
  1
 1 2
2
 3 1  3  (1   )   2 (   )

 
1      1      (1   ) 2  2 
3
 ( 2   2 )  3  3   3 
 
 4
2
 2   1
 2   14
2 
 1

 3  2   1  3  
1    1   3   1  
1
1
7  1
  (1   )     3  3    3  3    2  5  5   3 


3
 2  2  1
 2  1  2 1  2
2 13
2  23 
 2  1  2   1

 9   1     3 2
 9(1   ) 9 
9 
3 9(1   )

2
 A2  1 
 3  5  ( 2   2 )  3 
  (   )   
3
3
4 1
4 2 4 2 
2 2
2 1


 2
 4


9 2 2 
1
1  3   1   9 2   1  
 21  1
 6    1    A2    5  3  3    3 
   5 5 
3 
22 2 
2 

 4   2  2 1  2  1 2  2  4  1

3 

1
1
(3  2) (1   )  
  5 3 3 




2   2  2 1  2 1  23
2 13
2  23  

In order to find the collinear points, we put   0 in equation (4.56). Their abscissae are
the roots of the equation
g     
(1   )(   )  (  1   ) 3   (  1   )  1 
 1
 1 3

 A2  
  2  3 1   (1   )   
3
3
5
1
2
2 
2
 3
 2
 cd 
3  (1   )(   )  (  1   )   1    
(1   ) 2 (   )  2 (  1   )
 2 


3




 

2 
13
 23
14
 24
  1  2 
 7  1 1  7  (   ) (  1   )  (   ) (  1   )  3  2  (   )
  (1   )        



 3
13
 23
1 23
2 13
 1  2
 2  1  2  2 

1  3  (  1   )   A
2
3
2


2
 9   1    
3 3  9 (  1   ) 9(1   )(   ) 9  (  1   )  2


 1 
     

3
2
4  25
4 13
4  23
 4 



 7 21  1 1  7  3(  1   ) 
 3 9 9(1   ) 
2 2 (  1   ) 
 3 



   1    A2  3       


6
2 1 
2
2  25
 22 22


 4  2 4  1  2  2 

21  (   ) (  1   )  3 (  1   ) 3(   ) 3(  1   ) 3(3  2)(   )
 





4 
13
 23
2  25
2 13  2
2 1  23
4 13


3(1   )(  1   ) 3 (  1   ) 

  0
4  23
2  23

(4.57)
with 1     ,  2    1   .
105
To locate the collinear points on the   axis, the orbital plane is divided into three
parts:   1 ,1     2 and  2   with respect to the primaries where 1   and
 2  1 
m1  1  
(a)
1 ,0   ,0
m2  
1
 2 ,0  1  ,0
C
m1  1  
L2
L1
 ,0
2
m2  
(b)
1 ,0   ,0
L3
(c)
 ,0
1  3
 ,0
C
 2 ,0  1  ,0
m2  
m1  1  
1 ,0   ,0
C
 2 ,0  1  ,0
Figure 4.1: Reference parameter for collinear Lagrangian points
Case 1:
Position of L1    2  (see Fig. 4.1 (a))
Let   2  1;   1  1  1    1  1  1; since the distance between the primaries
is unity, i.e.  2  1  1  1   and  2  1   then   1  1  ; 1  1  1; 2  1
with  i  0i  1,2
(4.58)
Now substituting equation (4.58) in (4.57), we obtain
106
 6 A2  2  111   36 A2  6  18 A2   12  110   26  4cd 2  6 A2cd 2  18  87 A2   81A2  15 A2  2  2 2  19
  6 A2cd 2   2  4 2  3 A2  3  32  24 A2cd 2  159 A2   93 A2  16cd 2  2  3  4  cd 2  57 A2  2  18
  12 cd 2  81A2  2  24  18 A2cd 2   135 A2   36 A2cd 2  6  3  9 A2  3  63 A2  24cd 2  17
  32  24 A2cd 2  9 A2  3  36 A2  2  12 cd 2  27 A2  12cd 2  14  2  18 A2cd 2   6  3  18  39 A2   16
  6 A2cd 2  46 A2   28 2  55 A2  2  6 A2  6 3  3 A2  3  12 cd 2  32  6 A2cd 2   15
  30  6 A2cd 2   15 A2  3  120 A2  2  96 A2   6 3  12 cd 2  24  2  14
  112 A2  2  18 A2cd 2   10  21A2  3  4 cd 2  96 A2   2 3  13
  52 A2   46 A2  2  18 A2  3  18 A2cd 2   4 2  12   6 A2  2  12 A2   6 A2cd 2   6 A2  3  1  8 A2  2  0
(4.59)
In the presence of oblateness effect only, the corresponding equation is
 3 A2  2  17   9 A2  6  3 A2   2  16   9 A2  6 A2   4  6  15  3 A2  3 A2   2  14
413   3 A2   2  12  6 A2 1  3 A2   0
(4.59a)
Case2: Position of L2 1     2  (see Fig. 4.1 (b))
Let 2    2 ;   1  1  2    1  2  ; 1  1  2 ;2  2 with  i  0i  1,2
(4.60)
Substituting equation (4.60) in (4.57), the following equation is obtained
 6 A2  2  211   36 A2  6  18 A2   12  210   4cd 2  26  6 A2cd 2  87 A2   18  81A2  2  2  15 A2  2  29
  4 2  3 A2  3  4 cd 2  24 A2cd 2  177 A2   93 A2  44   2  57 A2  2  2  3  16cd 2  6 A2cd 2   28
  18 A2 cd 2   9 A2  3  189 A2   6  3  36   12  cd 2  81A2  2  36 A2 cd 2  24cd 2  63 A2  24  27
 18 A2 cd 2   2 2  105 A2   60 A2  2  18  15 A2  3  4  cd 2  12cd 2  2  3  16   24 A2cd 2  27 A2  2 6
  6 A2  15 A2  3  36 2  6 A2cd 2   6 3  32  64 A2   12  cd 2  73 A2  2  6 A2cd 2  25
  48 2  6 A2 cd 2   6  3  30   96 A2   12  cd 2  120 A2  2  15 A2  3  2 4
  10  128 A2  2  96 A2   21A2  3  18 A2cd 2   2 3  24 2  4 cd 2  23
  18 A2 cd 2   52 A2   18 A2  3  94 A2  2  4  2  2 2   12 A2   6 A2 cd 2   42 A2  2  6 A2  3  2  8 A2  2  0
(4.61)
107
In the presence of oblateness effect only, the corresponding equation is
 3 A2  272  9 A2  6  3 A2   2 62   9 A2  6 A2   4  652  3 A2  3 A2   2 42
 432  3 A2   2 22  6 A2 2  3 A2   0
(4.61a)
Case 3: Position of L3   1  (see Fig. 4.1 (c))
Let the distance of the point L3 from the bigger primary be 1  3
2  1  1  1    1  3 ;2    2  3 and   3    1; 1  1  3 ; 2  2  3 with
 i  0i  1,2
(4.62)
substituting equation (4.62) in (4.57), the following equation is obtained
 6 A2  2  311   96 A2  18 A2   32  6  310   6 A2cd 2  681A2  218  94  273 A2   15 A2  2
4cd 2  2 2  39   213 A2  2  32 2  6 A2cd 2   3 A2  3  84 A2cd 2  818  644   4  cd 2  2823 A2
 56cd 2  1833 A2   2 3  38   52 cd 2  2528  224  2  344cd 2  26  3  1796  1329 A2  2  516 A2cd 2
  7593 A2  78 A2cd 2   7161A2  39 A2  3  37   13929 A2  438 A2cd 2   1212cd 2  2126  146  3
292 cd 2  890 2  1824 A2cd 2  17997 A2  6260   4764 A2  2  219 A2  3  36  524  466  3  17868 A2
1386 A2cd 2   932 cd 2  687 A2  3  2680cd 2  10148  2192 2  10771A2  2  30374 A2   4086 A2cd 2  35
  934 3  10814   2224  16212 A2  3808cd 2  2694 A2cd 2   3440  2  1305 A2  3  34916 A2  1868 c 2
15826 A2  2  6012 A2cd 2  34   3408  210416 A2  3424  1515 A2  3  14940 A2  2  5808 A2cd 2 1218  3
3282 A2  cd 2  2436 cd 2  3392cd 2 7374   27212 A2   33   14060 A2   4656 A2  2040cd 2   2044  2
2208  1020 A2  3 3552 A2cd 2  1728cd 2  8554 A2  2  1020  3  2442 A2cd 2   2948   32   688  2
1014 A2cd 2   504 3  576  4532 A2   1248 A2cd 2  2578 A2  2  342 A2  3  1344 A2  384cd 2  488
1008 cd 2  3   192 A2cd 2  276 A2  2  48  112  2  192 A2  36 A2  3  224  cd 2  180 A2  cd 2
728 A2   112 3   0
(4.63)
108
In the presence of oblateness effect only, the corresponding equation is
 3 A2  2  37   33 A2  3 A2   2  22  36  153 A2  30 A2   20  102  35   387 A2  123 A2 
82 256  34   264 A2   576 A2  180  368 33   309 A2   504 A2  288  230  32  186 A2 
240 A2  168  96  3  45 A2   56  48 A2  0
(4.63a)
It is noticed that in each case there exists only one physically reasonable root.
4.4.3 Stability of collinear points
The stability of an equilibrium configuration is examined; that is, its ability to restrain
the body motion in its vicinity. To do so the infinitesimal body is displaced a little from
an equilibrium point with a small velocity. If its motion is rapid departure from vicinity
of the point, we call such a position of equilibrium an unstable one. If the body
oscillates about the point, it is said to be a stable position.
In order to study the stability of the collinear points, we consider the characteristic
equation given by
( p1q2  p2 q1 ) 4  ( p1q6  p5q2  p3q4  p6q1  p2q5  p4q3 ) 2  p5q6  p6q5  0
(4.64)
where,
p1  1  W0 , p2  W0 , p3  W0  W0  0, p4  W0  2nd  W0 ,
p5  W0 , p6  W0 ,
0
0
0
q1  W0 , q2  1  W
  , q3  2nd  W  W ,
q4  W0  W0  0, q5  W0 , q6  W0 .
The second order partial derivative of W are denoted by subscripts. The superscript 0
109
indicates that the derivative is to be evaluated at the collinear equilibrium points
 0 , 0  under consideration.
In order to study the stability of the collinear points it requires to study the motion in the
proximity of these points, hence in this case, the second order derivatives evaluated at
0 , 0  are
W0  1 
 3 15 (  1   ) 2 3 
3(1   )(   ) 2 (1   ) 3 (  1   ) 2 




A
 5
2 
15
13
 25
 23
2  27
22 
2

1
3 2  9(1   )(   ) 2 3(1   )

3


(1


)

 


cd 2
2
2 15
2 13


9 (  1   ) 2 3  2  3(1   )(   ) 3 (  1   )  3(1   ) 3
 3   2 


 
2  25
22 
13
 23
1
2



 7(   ) 7(  1   )
4(1   ) 2 (   ) 2 (1   ) 2 4  2 (  1   ) 2  2


 4   (1   ) 

6
4
6
1
1
2
2
13
 23

7  3(   ) 2 1 3(  1   ) 2 1  3(   ) 2 2(   )(  1   )
1
3(  1   ) 2
 








2  15
13
 25
 23 
15  2
13  23
13  2
1  25
1
3(3  2)(   ) 2 (3  2) 3(1  3 )(  1   ) 2 (1  3 )
9
3





 A2    1     3   2
3
5
3
5
3
1  2
2 1
2 1
22
22
2
4
3  9(1   )(   ) 2 3(1   ) 9  (  1   ) 2 3   45 (  1   ) 2 9   2
 2 


 3 
 5 
4 
15
13
 25
2  
4  27
42 
 3(1   )(   ) 3 (  1   )  9 (  1   )  9(1   ) 9  3  16  2 (  1   ) 2 2  2 
3  

 


 6 


13
 23
 25
2  2 2  23 
 28
 2 


 2 1
2
3  21  3(   ) 2 1
 21    1     1       7  15(  1   )
  1    A2  





 3


  
5
 23
13  2 
2  27
2  25  4  15
1
 2   2

1 3(  1   ) 2  15 (  1   ) 2 3 9(   ) 2 3(   )(  1   )
3
3
9(  1   ) 2









 23
 25
2  27
2  25
2 15  2
13  23
2 13  2 2 1  23
2 1  25


9(3  2)(   ) 2 3(3  2) 3(1  3 ) 9(1  3 )(  1   ) 2 




4 15
4 13
4  23
4  25

(4.65)
110
W0  1 

 3 3 
(1   ) 
 3  A2   5 
3
1
2
 2 22 
 3(1   ) 3  2 3(1   ) 3 (1   ) 2 3
1 
3
3   (1   )   2  
 3  
 
 4
2 
3
cd 
2
2 2 
1
2
12
2
 2 1
1
1
(3  2) (1  3 ) 
 7  1 1   (1   )
  (1   )    3  3   3  3  3 



3
2
1 2 1  2
2 13
2  23 
 2  1 2  1
3
3  3(1   ) 3  9  2 9(1   ) 9  3 2  2 
3
 A2    1     3   2   2  
 3  5  




2
2  2 13
2 2  4 2
2 1
2  2 2  23  26 
4
 21 21  1 1  3 3(1   ) 3
3
3
3(3  2) 3(1  3 ) 
  1    A2  5 
 5 3 



 3  3  3 
3
3
22
2  2 2 1  2 2 1  2
4 13
4  23 
 4  2 4  1 2  2 1
(4.66)
W0  0
(4.67)
W0 
1
cd 2
 1 2 3(1   ) 3
3
3  

 A2   2  3  
  
1
2
2 2 
4
2
(4.68)
0
W
 
1
cd 2
 3 2 3(1   ) 3
9
3  

 A2   2  3  
  
1
2
22  
4
2
(4.69)
W0  0
(4.70)
W0  0
(4.71)
W0  0
(4.72)
It will be shown that, the discriminant  of (4.64) is positive at the collinear points
Li i  1,2,3
To show  is positive it is suffices to notice that
111
M  4( p1 q 2  p 2 q1 )( p5 q 6  p 6 q5 )  0
(4.73)
as shown below
M can also be written as


M  4W0 W0 1  W0 1  W0

(4.74)


From (4.68) and (4.69) it is clear that 1  W0  0 and 1  W0   0
Now, the signs of W0 and W0 are studied at the collinear points Li i  1,2,3
Firstly, we will do this at L1 , since the coordinates of this point is 1  1   , 0  , then
1  1  1 and 2  1 where 0  1  1, hence we can write W0 and W0 as a function
of 1 , say h(1 ) and f (1 ), respectively. In this case from (4.65), h(1 )  h(0 )  
and from (4.66), f (1 )  f (0 )  , hence W0  0 and W0  0 contrary to the
classical case where W0  0 and W0  0. However W0 W0  0 and consequently
M  0. Hence the discriminant of the equation (4.64) is positive, and the characteristic
roots can be written as 1,2   , 3,4  i where  and  are real.
Thus 1,2 are real and 3,4 are pure imaginary, hence the motion around the collinear
point L1 , is unbounded and the solution is unstable.
Similarly, it can be shown that the points L2 , L3 are also unstable.
112
CHAPTER FIVE
RESULTS AND DISCUSSIONS
5.1 Introduction
In this chapter the effects of oblateness, triaxiality and radiation of the primaries as well
as the effects of perturbations in the Coriolis and centrifugal forces on the locations and
stability of the triangular points are discussed. The results are also validated with the
classical results and other existing results from the various aspects of generalizations.
5.2 Results and Discussion of Section 3.2
The triangular libration points in the relativistic restricted three-body problem, under the
assumption that the primaries are luminous and triaxial are discussed here. The
positions of the triangular points in equation (3.15) are obtained. It can be seen that they
are affected by the relativistic, radiation and triaxiality factors. It is important to note
that these triangular libration points cease to be classical one that is they no longer form
equilateral triangles with the primaries. Rather they form scalene triangles with the
primaries. Equation (3.42) gives the critical value of the mass parameter c of the
system which depends upon relativistic factor, triaxiality parameters  i ,  i
(i  1,2)
and radiation factors  i (i  1,2) . In the absence of relativistic factor, the results
obtained in this study are in agreement with those of Sharma et al. (2001b), Singh
(2013) when there is no perturbations in the Coriolis and centrifugal forces
i.e.      0. When the primaries are non triaxial.
It is noticed from equation (3.44) that that radiation and triaxiality both have
destabilizing effects, and therefore the size of the range of stability decreases with
increase of the values of these parameters. Evidently, it can also be seen that the
113
relativistic factor reduces the size of stability region. When the primaries are nonluminous and non-triaxial, the stability results obtained in this study are in accordance
with those of Douskos and Perdios (2002) and disagree with Bhatnagar and Hallan
(1998). In the absence of relativistic factor, the results obtained in this study are in
agreement with those of Sharma et al. (2001a) and those of Singh (2013) when there is
no perturbations in the Coriolis and centrifugal forces (i.e.      0 ) in his study.
When the primaries are oblate spheroids (i.e.  1   2 ,  1   2 ) , the results of
equation (3.15) in this study differ from those of Katour et al. (2014) when the radiation
terms are not included in the relativistic part of the potential W in their study.
By considering the primaries as triaxial rigid bodies and sources of radiation in the
relativistic CR3BP, the positions of the triangular points are determined and their linear
stability is studied. It is found that their positions and stability region are affected by
relativistic, triaxiality and radiation factors. It is further observed that the relativistic,
triaxiality and radiation factors have destabilizing tendencies resulting in a decrease in
the size of the region of stability. It is noticed that the expressions for A, D, A2 , C2 in
Bhatnagar and Hallan (1998) differ from the present study when the radiation pressure
factors are absent and the primaries are spherical (i.e. i   i   i  0, i  1, 2) ; keeping
in mind that the expressions denoted by A1 and A2 in Bhatnagar and Hallan (1998) are
denoted in this study by E and F respectively. Consequently, the characteristic
equation is also different. This made Bhatnagar and Hallan (1998) to infer that the
triangular points are unstable, contrary to Douskos and Perdios (2002) and our results.
The present results are also in disagreement with those of Katour et al. (2014). One
major distinction is that the expression of the mean motion which they used in their
study differ from our own. It seems that there is an error in their expression.
114
5.3 Results and Discussion of Section 3.3
The triangular equilibrium points in the relativistic R3BP are dicussed under the
assumption that the bigger primary is a triaxial rigid body and the smaller one an oblate
spheroid.
In analogy to corresponding problem without oblateness and triaxiality factors, the
positions of analogous triangular equilibrium points in equation (3.52) are obtained. It is
seen from equation (3.52) that these triangular points are affected by the relativistic,
triaxiality and oblateness coefficients. It is important to note that these equilibrium
points in equation (3.52) cease to be classical i.e. they no longer form equilateral
triangles with the primaries because they do not meet 1   2  1.
Rather they form
scalene triangles with primaries.
Equation (3.67) gives the critical value of the mass parameter c of the system which
depends upon relativistic, triaxiality and oblateness factors. This critical value is used to
determine the size of the region of stability of the triangular points and also helps in
analyzing the behavior of the parameters involved therein. It is remarkable from
equation (3.68) that these parameters reduce the size of the stability.
In the absence of triaxiality and oblateness factors (i.e. A2  1   2  0, ), the stability
results obtained in this study are in accordance with those of Douskos and Perdios
(2002) and disagree with those of Bhatnagar and Hallan (1998) and Ahmed et al.
(2006). In the absence of relativistic terms, the present results are in accordance with of
Singh and Umar (2014) when the semi-major axis is unity in the absence of eccentricity
(i.e. a  1, e  0).
115
However when the bigger primary is an oblate spheroid (i.e. 1   2  A1 ), there are
apparent disagreements with the results of the present study with those of Katour et al.
(2014) when the primaries are non-luminous.
By considering the bigger primary as a triaxial rigid body and the smaller one an oblate
spheroid in the relativistic R3BP, the positions of the triangular points are obtained and
their linear stability is studied. It is discovered that their positions and stability are
affected by the relativistic, oblateness and triaxiality factors. It is also noticed that these
factors reduce the size of stability region.
It is also noticed that the expressions for A, D, A2 , C2 in Bhatnagar and Hallan (1998)
differ from the present study in the absence of oblateness and triaxiality factors
(i.e. A2  1   2  0, ) ; keeping in mind that the expressions denoted by A1 and A2 in
Bhatnagar and Hallan (1998) are denoted in this study by E and F respectively.
Consequently the expressions p1 , p3 , p4 , p5 and the characteristic equation are also
different. This led them (Bhatnagar and Hallan, 1998) to infer that the triangular points
are unstable, contrary to Douskos and Perdios (2002) and the present results.
Also, the result of the present study, when the bigger primary is oblate
(i.e.  1   2  A1 ) , differ from those of Katour et al. (2014) when the primaries are
non-luminous. Our major distinction is that the expression of the perturbed mean
motion equation (3.47) in this study differ from their own and consequently leads to
different results. In addition to that, we have studied the stability of the dynamical
system while they have not. It seems that there is an error in the expression of the
perturbed mean motion which they have used.
116
5.4 Results and Discussion of Section 3.4
It can also be observed from equation (3.73) that the frequencies of the orbits of the
long and short periodic motion are dependent on all the parameters involved. From
equation (3.74), one can infer that the periodic orbits are elliptical. Also from equation
(3.80) it can be observed that the orientation of these orbits are affected by triaxiality
and relativistic factors. The eccentricities of the long and short periodic orbits are
obtained in equation (3.86). It is seen that they are also affected by the triaxiality and
relativistic factors.
Equations (3.88) and (3.89) provide the semi-major and semi-minor axes of the long
and short periods. In the absence of relativistic and triaxiality factors i.e. when
1
0
cd 2
and 1   2  0 , the results of the present study correspond to those of the classical case
(Szebehely,1967a) and the results of equations (3.73) and (3.80) coincide with those of
Abouelmagd and El- Shaboury (2012) when the oblateness and radiation factors are
absent. In the absence of relativistic factor our results also validate those of Sharma et
al. (2001b) when only the bigger primary is considered triaxial and  tends to zero in
their study.
The periodic orbit in the vicinity of the triangular points for 0    c have been
investigated. The frequencies, the coefficients of long and short periodic terms, the
elliptic orbits, the orientation, the lengths of semi-major and semi-minor axes have also
been found. They are all affected by the triaxiality and relativistic factors.
117
5. 5 Results and Discussion of Section 3.5
5.5.1 Numerical results
The equations (3.96), (3.98), (3.100) are used respectively to compute the positions of
the collinear points L1 , L2 , L3 in the presence of relativistic and triaxiality factors while
equations (3.96a), (3.98a), (3.100a) are used to compute their positions in the presence
of triaxiality factors only. For the numerical application, we use the Sun-Earth system,
Earth-Moon system and Sun-Pluto system. Five different cases of different set of semiaxes in km. (h, b, f ) of the smaller primary are taken. For the Sun-Earth system, we take
(6400, 6400, 6400), (6400, 6390, 6380), (6400, 6380, 6360), (6400, 6370, 6340) and
(6400, 6360, 6320), for the Earth –Moon system, (1738,1738,1738), (1738,1728,1718),
(1738,1718,1698), (1738,1708,1678),(1738,1698,1658) and for the Sun-Pluto system,
(3000,3000,3000),
(3000,2990,2980),
(3000,2980,2960),
(3000,2970,2940),
(3000,2960,2920). Some of the data has been borrowed from Ragos et al. (2001) and
Sharma and Subba Rao (1975). In Tables 5.1, 5.2, 5.3 these positions are presented. The
corresponding positions of the classical problem are also included. The corresponding
positions in the presence of triaxiality effect only (second entry in table for each system)
are
also
included.
118
Table 5.1: Sun-Earth system,   0.000003003500, cd
parameter
Classical
case1
 10064.84, R  149597870.61km
case 2
case 3
case 4
case 5
1
0
2.284  10
12
4.561  10
12
6.831  10
2
0
1.141  10
12
2.277  10
12
3.48  10
L1
L2
L3
1.0100341380 9074
0.99002657245077
 1.00000125145833
1.01003413806000
0.99002657248300
 1.00000125145831
12
12
9.094  10
12
4.533  10
12
1.01003413823000
1.01003413840000
1.01003413857000
1.01003413874000
1.01003413826000
1.01003413843000
1.01003413860000
1.01003413877000
0.99002657231200
0.99002657214000
0.99002657196900
0.99002657179900
0.99002657227900
0.99002657210800
0.99002657193700
0.99002657176600
 1.00000125145660
 1.00000125145489
 1.00000125145319
 1.00000125145148
 1.00000125145662
 1.00000125145491
 1.00000125145321
 1.00000125145151
119
Table 5.2: Sun-Pluto system,   0.00000006500, cd  63280.18, R  5900 106 km
parameter
Classical
case1
case 2
case 3
case 4
case 5
1
0
6.871588624 10
16
1.369721344 10
15
2.047687446 10
15
2.721057167 10
15
2
0
3.430048837 10
16
6.825624820 10
16
1.018672795 10
15
1.351335823 10
15
L1
L2
L3
1.00129454074324
0.9987 0656252876
 1.0000000270833
1.00129454074300
0.99870656252900
 1.0000000270833
1.00129454074400
1.00129454074400
1.00129454074400
1.00129454074500
1.00129454074400
1.00129454074400
1.00129454074400
1.00129454074500
0.99870656252800
0.99870656252800
0.99870656252800
0.99870656252700
0.99870656252800
0.99870656252800
0.99870656252800
0.99870656252700
 1.0000000270833
 1.0000000270833
 1.0000000270833
 1.0000000270833
 1.0000000270833
 1.0000000270833
 1.0000000270833
 1.0000000270833
120
Table 5.3: Earth-Moon system,   0.0121314293, cd
parameter
Classical
case1
case 2
case 3
1
0
9.375000000 10
2
0
4.673936632 10
L1
L2
L3
1.155608442501
0.837009426190
 1.00505466491
1.155608444240
0.837009426300
 1.005054664477
 292624.8185, R  384000km
8
8
1.864149306 10
case 4
case 5
7
2.77994791 10
7
3.684895833 10
7
8
1.37776692 10
7
1.820746528 10
7
9.266493056 10
1.155608847700
1.155609249000
1.155609646500
1.155610040000
1.155608847700
1.155609249000
1.155609646400
1.155610040000
0.837008972600
0.837008523300
0.837008078400
0.837007637900
0.837008972800
0.837008523600
0.837008078900
0.837007638500
 1.005054594295
 1.005054524786
 1.005054455958
 1.005054387803
 1.005054594296
 1.005054524791
 1.005054455960
 1.00505438780
121
5.5.2 Results and discussion
Equations (3.90)-(3.91) describe the motion of a third body under the influence of the
triaxiality of the smaller primary and relativistic effect. Equations (3.96), (3.98), (3.100) give
respective position of the collinear equilibrium points L1 , L2 , L3 which are affected by the
relativistic and triaxiality factors. Equations (3.96a), (3.98a), (3.100a) give their positions in
the presence of triaxiality only. It can be seen in section 3.5.3 that the relativistic and
triaxiality factors are unable to change the instability character of the collinear points. For the
Sun-Earth system in the absence of triaxiality  1   2  0 the numerical results of the
present study are in agreement with those of Ragos et al. (2001). This agreement can be
easily seen in Table 5.1 when the notations L1 and L2 are interchanged.
For the Sun-Earth system it can be observed for the Sun-Earth system from Table 5.1 that L1
moves towards the origin from the classical position due to relativistic effect only, whereas it
moves from the classical position in the direction of the positive  -axis. The triaxiality has
more shift than that of the joint effect.
L2 moves along the positive  -axis from the classical position due to relativistic effect only,
whereas it moves towards the origin from the classical case due to triaxiality alone or both.
L3 has a shift towards the origin due to triaxiality. It has also a shift towards the origin due to
the joint effect. This shift is almost same as that of triaxiality. The relativistic shift in
comparison with that of triaxiality is not remarkable.
For the Sun-Pluto system, it is observed from Table 5.2 that the point L1 has a very small
shift towards the smaller primary from the classical position due to relativistic effect. The
triaxiality shifts L1 away from the classical position. This shift is almost similar as that of
joint effect. The point L2 has a very small shift in the direction of the positive  -axis from
122
the classical position due to relativistic effect alone. The triaxiality shifts L2 towards the
origin from the classical position. The similar shift is also seen due to the joint effect. The
classical position of L3 does not change due to triaxiality or relativistic or both effects.
For the Earth –Moon system, it can be seen from Table 5.3 that L1 moves towards the origin
from the classical position due to relativistic effect . The triaxiality shifts it from the classical
position away from the origin. The similar shift is also observed due to the joint effect. L2
and L3 all move towards the origin from the classical position due to relativistic or triaxiality
or both effects.
By considering the smaller primary as triaxial rigid body, the positions of collinear points are
obtained and their linear stability is also examined. It is found that their positions are both
affected by the relativistic terms and triaxiality parameters. This is confirmed from the tables.
It is further observed that in spite of the introduction of relativistic and triaxiality coefficients,
the collinear points remain unstable. Numerical investigations on this model by considering
the Sun-Earth, Sun-Pluto and Earth-Moon systems have been performed to show the
relativistic and triaxiality effects on collinear points
5.6 Results and Discussion of Section 4.2
Equations (4.1)-(4.3) describe the motion of a third body under the influence of oblateness of
the bigger primary together with small perturbations in the Coriolis and centrifugal forces in
the relativistic R3BP. Equations (4.8) give the positions of triangular equilibrium points,
which are affected by the oblateness, relativistic factor and a small perturbation in the
centrifugal force, but not that of Coriolis force because equation (4.8) is independent of the
parameter  2 ; while equation (4.23) gives the critical value of the mass parameter c of the
123
system which depends upon small perturbations 1 ,  2 given in the centrifugal and Coriolis
forces, oblateness parameter A1 and relativistic factor.
The critical value is used to determine the size of the region of stability of the triangular
points and also helps in analyzing the behavior of the parameters involved therein. Equation
(4.24) describes the region of stability. It is obvious from equation (4.24) that the relativistic
term, oblateness coefficient and a small perturbation  1  0 , in the centrifugal force all shrink
the stability region independently; whereas the small perturbation in the centrifugal force
expands it for  1  0 and that of the Coriolis force expands it for  2  0 and shrinks it for
 2  0. This can be explained by the presence of negative coefficients of the formers and
positive coefficient of the latter.
Even on considering the coupling terms
i
A1
and
 i  1, 2  which are very small
2
cd
cd 2
quantities, from mathematical points of view it can be observed from equation (4.24) that the
joint effect of relativistic and oblateness and that of relativistic and a small perturbation  1  0
in the centrifugal force expand the size of region of stability; whereas the joint effect of
relativistic and a small perturbation  2 in Coriolis force reduces it for  2  0 and expands it for
 2  0. Similarly, the joint effect of relativistic term and a small perturbation  1 in the
centrifugal force reduces it for  1  0. This is also as a result of the positive coefficients of the
coupling terms
A1


and 12 and negative coefficient of the coupling term 22 . However, the
2
cd
cd
cd
net effect is that the size of region of stability increases or decreases or remains unchanged
according as p  0 or p  0 or p  0, respectively. In the absence of perturbations and
oblateness  i  A1  0, i  1,2, the results of the present study are in agreement with those of
Douskos and Perdios (2002) and disagree with those of Bhatnagar and Hallan (1998).
124
 1

In the absence of relativistic terms and centrifugal force  2  0, 1  0  ,
 cd

the results
coincide with those of Szebehely (1967b). In the absence of relativistic terms and oblateness
 1

 2  0, A1  0  , our results are in agreement with those of Bhatnagar and Hallan (1978).
 cd

 1

In the absence of relativistic terms and perturbations  2  0,  i  0, i  1, 2  , the results of
 cd

the present study coincide with those of SubbaRao and Sharma (1975). In the absence of
relativistic terms, our results are in accordance with those of Abouelmagd et al. (2013) when
the bigger primary is oblate and the mixed effect A1 i i  1,2 is ignored in their study.
By considering the bigger primary as an oblate spheroid body under the influence of small
perturbations in the Coriolis and centrifugal forces in the relativistic R3BP, the positions of
the triangular points are determined and their linear stability is also investigated. It is found
that the effect of relativistic terms, oblateness and a small change in the centrifugal force on
these positions are quite prominent. It may also be seen that relativistic terms, oblateness and
a small change in the centrifugal force all reduce the size of region of stability independently,
where a small perturbation in the Coriolis force expands it.
We have observed the expressions for A, D, A2 , C2 in Bhatnagar and Hallan (1998) differ
from the present study when the oblateness and small perturbations in the Coriolis and
centrifugal forces are absent i.e.  i  A1  0, i  1,2 ; bearing in mind that the expressions
denoted by A1 and A2 in Bhatnagar and Hallan (1998) are denoted in this study by E and F
respectively. Consequently, the expressions p1 , p3 , p4 , p5 and the characteristic equation are
also different. This led Bhatnagar and Hallan (1998) to infer that the triangular points are
unstable, contrary to Douskos and Perdios (2002) and the present results.
125
5.7 Results and Discussion of Section 4.3
The triangular libration points in the perturbed relativistic R3BP under the assumption that
the bigger primary is a triaxial rigid body are discussed. In analogy to corresponding problem
without perturbations and triaxiality, the positions of analogous triangular libration points in
equation (4.33) are obtained. It is important to note that these triangular libration points in
equation (4.33) cease to be classical ones i.e. they no longer form equilateral triangles with
the primaries as they do in the classical case. Rather they form scalene triangles with the
primaries. It is seen from equation (4.33), that the positions of triangular points are affected
by the relativistic effect, triaxiality and the perturbation in the centrifugal force, but not
affected by that of the Coriolis force.
Equation (4.48) gives the critical value of the mass parameter c of the system which depends
upon triaxiality, relativistic factor and small perturbations 1,  2 in the centrifugal and Coriolis
forces, respectively. This critical value determines the size of the region of stability of the
triangular points and also helps in analyzing the behavior of the parameters involved therein.
It is obvious from equation (4.50) that the relativistic and triaxiality effects reduce the size of
the stability region separately where as the Coriolis effect expands it if  2  0 and shrinks it if
 2  0. Similarly the separate effect of centrifugal force expands it if  1  0 and shrinks it if
 1  0.
Even on considering the coupling terms
i
cd 2
 i  1, 2 
which are very small quantities, from
mathematical point of view, it can be seen that from equation (4.50) that the joint effect of the
relativistic term and centrifugal force that is the term containing the coupling term
1
cd 2
expands the size of the stability region if 1  0 and shrinks it if 1  0 ; whereas the joint
126
effect of the relativistic and Coriolis force i.e. the term containing the coupling term
2
cd 2
shrinks it if  2  0 and expands it if  2  0.
From the overall analysis, it is clear that the Coriolis and centrifugal forces maintained their
stabilizing and destabilizing characteristic behavior respectively.
However, it can be seen that from equation (4.51) that the net effect is that the size of the
range of stability increases or decreases according as p  0 or p  0 where p depends upon
relativistic, centrifugal and Coriolis effects.
In the absence of relativistic terms, our result coincides with those of Singh (2013) when the
primaries are non-luminous and only the bigger primary is triaxial . In the absence of small
perturbations and the bigger primary is oblate (i.e. 1   2  A1 ), the result of equation (4.33)
are in disagreement with those of Katour et al. (2014) when the primaries are non-luminous
and smaller one spherical.
In the absence of relativistic effect and of small perturbations i.e. 1   2  0, the results
obtained in this study are in agreement with those of Sharma et al. (2001a) when the bigger
primary is triaxial only; and those of Sharma et al. (2001b) when the primaries are nonluminous and the bigger one is triaxial only.
Under the assumption that the bigger primary is a triaxial rigid body and small perturbations
 1 ,  2 are given to the centrifugal and Coriolis forces, the stability of the triangular
equilibrium points in the relativistic R3BP has been examined. It is found that their positions
are affected by the relativistic factor, a small change in the centrifugal force and triaxiality
factors of the bigger primary. It is also observable from equation (4.48) that all the
127
parameters involved in this study except the Coriolis force have destabilizing tendencies
resulting in a decrease in the size of the region of stability.
It is also noticed that the expressions for A, D, A2 , C2 in Bhatnagar and Hallan (1998) differ
from the present unperturbed study; bearing in mind that the expressions denoted by A1 and
A2 in Bhatnagar and Hallan (1998) are denoted in this study by E and F respectively.
Consequently, the expressions p1 , p3 , p4 , p5 and the characteristic equation are also different.
This led Bhatnagar and Hallan (1998) to infer that the triangular points are unstable, contrary
to Douskos and Perdios (2002) and our results.
There are striking differences between results obtained from equation (4.33) when the bigger
primary is oblate  i.e.1   2  A1  and perturbations absent and those of Katour et al. (2014)
when the primaries are non-luminous and only the bigger one is oblate. The reason is that the
perturbed mean motion in equation (4.28) used in this study differs from their own. It seems
that there is an error in the perturbed mean motion which they have used.
A practical application of this model could be the study of the motion of a dust grain particle
near Pluto and its moon Charon.
5.8 Results and Discussion of Section 4.4
5.8.1 Numerical results
The necessary data used have been borrowed from Sharma and SubbaRao (1975) and Ragos
et al. (2001). Some members of the solar system are used (mentioned in Table 5.4) to
examine the existence and position of the collinear equilibrium points. Equations (4.59),
(4.61), (4.63) and (4.59a), (4.61a), (4.63a) have been solved for the various pairs of the solar
system. In Table 5.5 the positions of collinear points of the Sun-Planet pairs are presented.
The corresponding positions respectively in the classical problem, classical problem with
128
oblateness, relativistic problem and relativistic problem with oblateness as included for
comparison purposes (first entry, second entry, third entry and forth entry respectively for
each system).
Table 5.4: Parameters of the systems

S.No
System
1
Sun-Earth
10064.84
0.000003003500 0.0000000007 108
2
Sun-Mars
12424.24
0.000000322700 0.0000000001 108
3
Sun-Jupiter
22947.35
0.000953692200 0.0000192887 108
4
Sun-Saturn
31050.90
0.000285726000 0.0000018690 108
5
Sun-Uranus
44056.13
0.000043548000 0.0000000070 108
6
Sun-Neptune
55148.85
0.000051668900 0.0000000010 108
cd
129
A2
Table 5.5: Positions of the collinear equilibrium points
S/No of the
L1
L2
L3
1.01003413809074
0.99002657245074
-1.00000125145833
1.01003413809000
0.99002657245100
-1.00000125145833
1.01003413806000
0.99002657248500
-1.00000125145831
1.01003413806000
0.99002657248500
-1.00000125145831
1.00476303037278
0.99525140276082
-1.00000013445833
1.00476303037300
0.99525140276100
-1.00000013445833
1.00476335306300
0.99525140276100
-1.00000013445833
1.00476335306300
0.99525140276100
-1.00000013445833
1.06882613997466
0.93236993769216
-1.00039737170283
1.06882613998000
0.93236993769000
-1.00039737170270
1.06882613992000
0.93236993764000
-1.00039737170120
1.06882613992000
0.93236993764000
-1.00039737170120
1.04606932684648
0.95474919731454
-1.00011905249873
1.04606932685000
0.95474919731000
-1.00011905249870
1.04606932683000
0.95474921600000
-1.00011905249850
1.04606932683000
0.95474921600000
-1.00011905249850
1.02454737494085
0.97576220621890
-1.00001814499999
1.02454668140000
0.97576220622000
-1.00001814499999
1.02454668140000
0.97576220665000
-1.00001814499998
1.02454668140000
0.97576220665000
-1.00001814499998
1.02599374139930
0.97434749094956
-1.00002152870832
1.02599374140000
0.97434749095000
-1.00002152870833
1.02599374140000
0.97434749156000
-1.00002152870831
1.02599374140000
0.97434749156000
-1.00002152870831
system
1
2
3
4
5
6
130
5.8.2 Results and discussion
Equations (4.52)-(4.55) describe the motion of a third body under the influence of the
oblateness of the smaller primary and relativistic terms. Equations (4.59), (4.61) and (4.63)
give respective positions of the collinear equilibrium points L1 , L2 , L3 in the presence of
relativistic and oblateness factors while equations (4.59a), (4.61a) and (4.63a) give their
positions in the presence of oblateness factor only. It can be seen in section 4.4.3 that the
relativistic and oblateness factors are unable to alter the instability behavior of the collinear
points. It can be observed when comparing first and second entries of each Sun-Planet pair
that the positions of L1 and L2 are affected by oblateness in the classical problem, while when
comparing first and third entries it can be said that they are affected by the presence of the
relativistic terms. It is also noticed that the oblateness effect on the position of the collinear
point L3 of the classical problem in most of the cases is negligible when comparing first and
second entries except for the Sun- Jupiter system where it has a little effect, while also when
comparing first and third entries it can be said that the relativistic terms have negligible effect
on the position of L3 except for the Sun- Jupiter and Sun- Saturn systems.
It is also noticed that the oblateness and relativistic factors have separately the same effect on
the position of L1 of the Sun-Uranus system and also have same separate effect on the
position of L1 of the Sun-Neptune system as shown from second and third entries of those
systems.
However in all cases it is found that the third and forth entries of the relativistic problem only
and relativistic problem with oblateness respectively are same up to forteen decimal places.
This indicates that in the presence of relativistic terms, the effect of oblateness does not show
131
physically on the positions of collinear points in our Solar system. It is also observed that all
the parameters involved have no effect on the position of L3 of the Sun -Mars system.
By considering the smaller primary as an oblate spheroid, the positions of the collinear points
and their linear stability in the relativistic R3BP are studeid. It is found that in spite of the
inclusion of relativistic and oblateness coefficients, the instability behavior of the collinear
points remains unaltered. A numerical survey of some members of the Sun-Planet pairs of
our solar system reveals that the positions of L1 and L2 are significantly affected by the
oblateness in the absence of relativistic factor and by also relativistic factor in the absence of
oblateness ; while the position of L3 is negligibly affected by oblateness and relativistic
factors in most of the cases and more specifically all the parameters involved have no effect
on the position of L3 of the Sun-Mars system. It is observed that the oblateness and
relativistic factors have same separate effect on the position of L1 of the Sun-Uranus system
they have also same effect on the position of L1 of the Sun-Uranus system. It is also noticed
that in the presence of relativistic terms, the effect of oblateness does not show physically in
our solar system as comparing third and forth entries of Table 5.5.
132
CHAPTER SIX
SUMMARY, CONCLUSION AND RECOMMENDATIONS
6.1 Summary
In this thesis, the effects of oblateness, radiation and triaxiality of the primaries on locations
and stability of the triangular points are first investigated. Firstly the case where both
primaries are triaxial and sources of radiation are considered and then the case where the
bigger primary is triaxial and the smaller one is oblate. The locations and stability of the
collinear points when the smaller primary is triaxial are also investigated. The analytical and
numerical studies in this connection with the Sun-Earth, Sun-Pluto and Earth-Moon systems
have been carried out.
The frequencies, eccentricities, semi-major and semi minor axes of the periodic orbits around
stable triangular points when the bigger primary is triaxial have been obtained and are found
to be affected by the relativistic terms and triaxiality.
Lastly, the locations and stability of the triangular points with perturbations in Coriolis and
centrifugal forces with the bigger primary as an oblate spheroid and the smaller one as
spherical and then with a triaxial bigger primary and spherical smaller primary have been
studied and also the locations and stability of collinear points when the smaller primary is
oblate are studied.
6.2 Conclusion
The equilibrium points of the relativistic R3BP when the primaries are non-spherical as well
as sources of radiation under the influence of small perturbations in the Coriolis and
centrifugal forces have been examined.
133
It was found that, the perturbation in the centrifugal force, radiation and asphericity of the
primaries and relativistic terms affect the locations of the triangular points. It was also
observed that the triangular points are stable for 0     c and unstable for  c   
1
where
2
 c is the critical mass value which depends on the combined effects of the parameters
involved. It was also found that oblateness, triaxiality, radiation pressure forces, centrifugal
force and relativistic terms have destabilizing tendencies, while the stabilizing behavior of the
Coriolis force remains unchanged. However, the net effect is that the range of stability region
increases or decreases according as p is positive or negative, where p depends upon
perturbations factors. It was also found that the stability behavior of the collinear points does
not change despite the inclusion of triaxiality or oblateness. Hence, they remain unstable.
The periodic orbits around the stable triangular equilibrium points in the range 0     c
when the bigger primary is triaxial were studied. It was found that the long and short periods,
the eccentricities, the semi-major and semi-minor axes and also their orientation were
affected by the parameters involved.
The results of the present study generalize the classical and previous results obtained by
previous researchers. That is, the previous results as in Douskos and Perdios (2002) can be
deduced from it. There are also disagreements with some of the previous authors‟ results such
as in Bhatnagar and Hallan (1998), Ahmed et al. (2006) and in Katour et al. (2014) with the
present results. It seems that there are some errors therein.
6.3 Recommendations
The results of this study may be applied in space exploration programme. In this research, the
stability of equilibrium points in the relativistic restricted three-body problem with constant
134
mass and perturbations has been investigated. For further research, the same problem with
variable mass can be investigated.
135
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